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1 翻轉線性代數 勘誤檔案 喻超凡喻超弘編著 喻超凡數位企業有限公司 版權所有 翻印必究

2 9 喻超凡 姚碩 林郁叢書第 章向量空間 習 題. 判別下列何者是 R 3 的子空間 大同資工 (a) {(x, y, z) x + z =, x,y,z R} (b) {(x, y, z) x = y = z,x,y,z R} (c) {(x, y, z) z = x + y, x,y,z R}. Determine whether or not the following are subspace of R 3 交大資工 (a) {(x,x,x 3 ) T x + x 3 =} (b) {(x,x,x 3 ) T x = x = x 3 } (c) {(x,x,x 3 ) T x 3 = x + x } (d) {(x,x,x 3 ) T x 3 = x + x } 3. V i are sets in the vector space R 3 V = {(x,x,x 3 ) x } V = {(x,x,x 3 ) x = x =x 3 } Whether V i are subspaces of the vectors space R 3. 雲科大資工 4. Let R be the set of real numbers. Let B be a nonzero matrix in R m n. Define H = { M + B M R m n } where the operators + and denote the matrix addition and the scalar multiplication, respectively. Is (H, +, ) a subspace of (R m n, +, )?Justifyyouranswer. 交大資工 5. Let V be the vector space of all functions f from R to R. Which of the following sets of functions are subspaces of V? Note that for given f and g in V and the scalar c we define (f + g)(x) =f(x)+g(x) and(cf)(x) =cf(x) (a) all f such that f(x )=f (x). (b) all f such that f() = f(). (c) all f such that f(3) = + f( 5). (d) all f such that f( ) =. (e) all f which are continuous. 交大資工

3 喻超凡 姚碩 林郁叢書第 章向量空間 因 (S Y )={ y,, y m, v,, v n } 為 W 的一組基底, 故為線性獨立, 則由 (3) 式可知 將 (4) 式代回 () 式可得 γ = γ = = γ m = δ = δ = = δ n = (4) k α i x i + i= n β j v j = (5) j= 又 (X S) ={ x,, x k, v,, v n } 為 U 的一組基底, 故為線性獨立, 則由 (5) 式可知 α = α = = α k = β = β = = β n = (6) 由 (4) (6) 兩式可知 (X S Y ) 為線性獨立 即 (X S Y ) 為 (U +W ) 的一組基底, 故 (c) 因此 dim (U + W )=k + n + m dim (U)+dim (W ) = k + n + m + n = dim (U + W )+n = dim (U + W )+dim (U W ) 即 dim (U + W )=dim (U)+dim (W ) dim (U W ) () 若 V = U W, 則 (U W )={}, 故 dim (U W )=, 因此 dim (U + W )=dim (U)+dim (W ) 4. Let V be a finite-dimensional vector space, and let U, U, and U 3 be its subspaces. Which of the following statements are true? () dim (U U )=dim (U )+dim (U ) () dim (U + U )=dim (U )+dim (U ) dim (U U ) (3) dim (U U U 3 )=dim (U )+dim (U )+dim (U 3 ) (4) dim (U + U + U 3 )=dim (U )+dim (U )+dim (U 3 ) dim (U U ) dim (U U 3 ) dim (U U 3 )+dim (U U U 3 ) 台大電機 C

4 8 喻超凡 姚碩 林郁叢書第 3 章線性變換 即 R (T ) span { T (x ), T(x ),,T(x n ) } () (3) 由 () () 式知 R (T )=span { T (x ),T(x ),,T(x n ) } 4. 証明 Sylvester 維度定理 (Sylvester dimension theorem) 設 V U 為同佈於 F 之兩向量空間, 且變換 T : V U 為線性變換, 若 V 為有限維度, 則 nullity (T )+rank (T )=dim (V ) 中山電機 証 設 dim (V )=n, 且令 S = { x, x,, x k } (k <n) 為 N (T ) 的一組基底, 即 nullity (T )=k, 因 N (T ) 為 V 的子空間, 故由第 97 頁定理. 知, 我們可將 S 擴張至 V 的一組基底 若 S = { x, x,, x k, x k+,, x n } S = { T (x k+ ),T(x k+ ),,T(x n ) } 是 R (T ) 的基底, 即 rank (T )=dim {R (T )} = n k, 則定理得証 (a) 先証 S 生成 R (T ): 因 {x, x,, x k } 為 N (T ) 的基底, 故 T (x i )= (i =,,,k), 由第 4 頁定理 3. 知 R (T ) = span { T (x ),,T(x k ),T(x k+ ),T(x k+ ),,T(x n ) } (b) 再証 S 為線性獨立 : = span { T (x k+ ),T(x k+ ),,T(x n ) } 設 S 的線性組合式為 c k+ T (x k+ )+c k+ T (x k+ )+ + c n T (x n )= ()

5 4 喻超凡 姚碩 林郁叢書第 3 章線性變換 7. Let T : P (R) P 3 (R); T [f(x)] = xf(x) +f (x). ( f (x) istheformal derivative of f(x) ) (a) Prove that T is a linear transformation. (b) Find bases for N (T )andr(t). (c) Determine whether T is one to one or onto. 交大電子 (a) 令 α R 且 f(x) g(x) P (R), 故 因 T [f(x)] = xf(x)+f (x) T [g(x)] = xg(x)+g (x) 故 T 為線性變換 T [αf(x)+g(x)] = x[αf(x)+g(x)] + [αf(x)+g(x)] = αxf(x)+xg(x)+αf (x)+g (x) = α[xf(x)+f (x)] + [xg(x)+g (x)] = αt [f(x)] + T [g(x)] (b) 令 f(x) =c + c x + c x N (T ), 且 c c c R, 故 T [f(x)] = xf(x)+f (x) =x(c + c x + c x )+(c +c x) = c +(c +c )x + c x + c x 3 = () 因 S = {,x,x,x 3 } 為 P 3 (R) 的標準基底, 即 S 為線性獨立的集合, 故由 () 式可得 c = c +c = () c = 解 () 式可得 c = c = c =, 故 f(x) =, 即 N (T )={}, 故 N (T ) 的基底為 又因 且 h(x) =a + a x + a x P (R) ; (a,a,a R) T [h(x)] = xh(x)+h (x)

6 3. 概論喻超凡 姚碩 林郁叢書 5 6. Let P = {ax + bx + c ; a, b, c R} and let T : P R be the linear transformation defined by T (P (x)) = P (x) dx (a) Find a basis for Ker (T ). (b) Find a basis for range (T ). 台師數學 (a) 設 P (x) =ax + bx + c N (T ), 則 T (P (x)) = 即 b = 3 a c, 故 (ax + bx + c) dx = a 3 + b + c = P (x) =ax + bx + c = ax +( 3 a c)x + c = a(x x)+c( x +) 3 即 Ker (T )=span {x x, x +} 3 故 Ker (T ) 的基底為 {x x, x +} 3 (b) 設 P (x) =ax + bx + c P, 故 T (P (x)) = 則 R (T )=R, 則 R (T ) 基底為 {} (ax + bx + c) dx = a 3 + b + c 7. Let T : P (R) M (R) be a linear transformation defined by [ ] f() f() T (f) = f() Find a basis for the range R (T )oft. 中原應數

7 96 喻超凡 姚碩 林郁叢書第 3 章線性變換 因 rank (A) =3, 故 R (A) =CS (A), 即 R (T )=span {+x, x+ x,x + x 3 } 再令 g(x) =a + a x + a x + a 3 x 3 N (T ), 故 T (g) =, 即 a a a = 令 a = c, 故可解得 a 3 = c a = c a = c, 則 因此 a 3 g(x) =c( x + x x 3 ) N (T )=span { x + x x 3 } () 令 α = {, +x, +x + x, +x + x + x 3 }, 為 P 3 的有序基底故 [p(x)] α = x x x 3, [T (p(x))] α = y y y 3 則 因此 M =[T ] α, 又 x 4 [T (p(x))] α =[T ] α [p(x)] α y 4 T () = + x = + ( + x)+ ( + x + x )+ ( + x + x + x 3 ) T (+x) =+x+x = + (+x)+ (+x+x )+ (+x+x +x 3 ) T ( + x + x )=+x +x + x 3 = + ( + x)+ ( + x + x )+ ( + x + x + x 3 ) T ( + x + x + x 3 )=+x +x +x 3 = + ( + x)+ ( + x + x )+ ( + x + x + x 3 ) 故 M =

8 3. 線性變換的矩陣表示法喻超凡 姚碩 林郁叢書 97 (3) 令 p(x) =a + a x + a x + a 3 x 3, 由 可 上式的特解為 上式的齊次解為 N (T ), 故 T (p(x)) = + x +x + x 3 a a a a 3 a a a a 3 = c p a a a a 3 = = + 則 p(x) =c( x + x x 3 )+(+x + x ) 3. Find the matrix representations, with respect to the standard basis {e, e }, for the following linear transformations from R to R. Reflection on the line y = mx. 雲科大電機 令 β = {e, e }, 其中 e =(, ) e =(, ), 而直線 y = mx 的方向向量 e t =(,m) 且 tan α = m ( 如圖 ),

9 4. 概論喻超凡 姚碩 林郁叢書 63 () 為內積 ; 設 x y z C, 且 α C (a) <αx +z, y >= (αx +z)ay = αxay +zay = α<x, y > + < z, y > (b) 因 A = A, 故 < x, y > = (xay )=xay T =(ya x ) T =(yax )=< y, x > (c) 因 det(a λi) =λ 3λ +=, 故 A 的特徵值為 λ = 3 ± 5, 由 Rayleigh quotient 定理可知 3 5 xx xax 3+ 5 故 < x, x >= x Ax 3 5 xx (d) 由 (a) (b) (c) 可知, 為內積運算 (3) 不為內積 令 xx [ ] [ ] [ ] A =, B =,C= <A+ C,B>= tr(a + C + B) =3 <A,B>+ <C,B>= tr(a + B)+tr(C + B) =3+=4 故 <A+ C,B> < A,B>+ <C,B> 3. View a random variable as a vector. Define the inner product for two vectors (random variables) X and Y as follows : <X,Y >= E[XY ] Verify that <X,Y >is indeed an inner product. (Hint : there are three conditions that you need to verify.) 清華通訊甲 解 設 Z 為隨機變數, α β R (a) <αx+ βz,y > = E[(αX + βz)y ]=E[αXY + βzy] = αe[xy ]+βe[zy ] = α<x,y >+β <Z,Y >

10 4. 概論喻超凡 姚碩 林郁叢書 33 故 B 在 W 上的投影向量為 且 B 到 W 的距離為 B Proj W B = B D = [ Proj W B = D = <D,D>= 3 ] 5. Consider the inner product space M 3 3 with the Frobenius inner product. Let 3 A = () Find a symmetric matrix B and a skew-symmetric matrix C (a matrix C is said to be skew-symmetric if C T = C) such that A = B + C. () Let V be the subspace of M 3 3 consisting of all 3 3 skew-symmetric matrices. Find the orthogonal projection of A onto V. 台大電機 D () 因 () 令 其中 B = A + AT B = A + AT A = A + AT + A AT = B + C 為對稱矩陣, C = A AT 為反對稱矩陣, 即 3 5 = A A T ; C = = V = {E E T = E, E M 3 3 } U = {D D T = D, D M 3 3 } 故 M 3 3 = V U, 又 Frobenius inner product 定義為 <D,E>= tr(e T D)

11 4. 概論喻超凡 姚碩 林郁叢書 39 u = v < v, u > < u, u > u = 7 = u 3 = v 3 < v 3, u > < u, u > u < v 3, u > < u, u > u = ( 4 7 ) ( 7 ) = 因此 w = u u = 7, w = u u = 7 9 w 3 = u 3 u 3 = 則 {w, w, w 3 } 為 CS (A) 中的一組正規化正交基底 (4) Proj CS(A) y = < y, w > w + < y, w > w + < y, w 3 > w = =

12 4. 概論喻超凡 姚碩 林郁叢書 37 可改寫成 Ax =, 其中 [ ] A =, x = x x x 3 令 W = N (A), 則 W = CS (A T ), 故 u 在 W = N (A) 上的正交投影為 Proj W (u) = u Proj W (u) x 4 = u Proj CS(A T ) (u) = u A T (AA T ) Au 4 = Find the orthogonal projection of u =(5, 6, 7, ) T on the solution space of homogeneous linear system [ ] x x x 3 x 4 [ ] = 中正電機 令 則 [ ] A = 5 6, u = 7 Proj N(A) (u) = u Proj (u) =u Proj (u) N(A) CS(A T ) = u A T (AA T ) Au =

13 338 喻超凡 姚碩 林郁叢書第 4 章內積空間 即 x min u =, 故 x min = u, 因此在 R (A ) 中, x min 為唯一滿足 Ax = b 的解 (3) 因此由 () 知 x min R (A ), 故 v F m, 使得 x min = A v, 且滿足 Ax min = A(A v)=(aa )v = b Given A = 3 and b = 4 7 (a) Find all solution of Ax = b. (b) Find the minimal solution of Ax = b. 台大電機 (a) [A b] = R ( ) R ( ) 3 GGGGGGGGGGGGGGA 令 x =[x x x 3 ] T, 故可得 { x +x x 3 =7 x 3x 3 = 令 x 3 = c x =3c x =7 5c, 因此 Ax = b 的通解為 x 7 5c 7 5 x = x = 3c = + c 3 c x 3 (b) 由 AA v = b 可得 上式可解得 v v v 3 v = 7 = 4 7

14 4. 概論喻超凡 姚碩 林郁叢書 365. 設向量 V =(,, ), V =(,, ), V 3 =(3,, ) 政大應數 (a) 由向量 V 和向量 V 所構成之平面 K 的方程式為何? (b) 向量 V 3 在平面 K 上之垂直投影向量為何? 3. Consider the vector space C[, ] with the inner product <f,g>= f(x)g(x) dx (a) Consider the subspace W = span {,x,x }. Find an orthonormal basis for W. (b) Find the orthogonal projection of x 3 onto the subspace W. 4. Given the inner product space C[, ] with inner product 5. <f,g>= f(x)g(x) dx 交大資科 Find the best least squares approximation to + x on [, ] by a function from the subspace S spanned by and 4x. 交大資工 x x =3 x =. Find the least squares solution. 雲科大工管 x = 6. Find the minimal solution of the following system with all steps listed. x +y + z =4 x y +z = x +5y =9 台大電機 D 7. Given an over-determined linear system of equation Ax = b with A an m n (m n) matrix and b an m vector, the least square solution x l,s is the m dimensional vector x that minimizes the vector norm (vector length) of Ax b. Please find the least- square solution for the following over-determined linear systems: (describe the method you used to obtain the solution) 中山資工 3 3 x 3 x = x

15 4. 概論喻超凡 姚碩 林郁叢書 (,, ), (,, ), 6 (,, ) 8. a = 5 b = (a) {, 3 t π } (b) Proj 3t (sin t) = U π. Proj S (v) = {. (a) U 的基底為, } { U 的基底為, 5 3 (b) 在 U 上與 v 最接近向量為 Proj U v = 在 U 上與 v 最接近向量為 v Proj U v = 6. (a) x y + z = (b)( 3, 3, 3 ) } 3. (a) {, 3 x, x x = x = 6. x min = (x 3 )} (b) 3 5 x 7. x = ( ) T

16 438 喻超凡 姚碩 林郁叢書第 A 章廣義反矩陣 (pseudoinverse) 因 故矩陣 A 的 pseudoinverses 為 4 A = = [ ] A + = 5 5 A = 5 [ 4 ] 3. Find the pseudoinverses of 中興電機 U =, V = [ ], and UV 3 (a) 先求 U 的 pseudoinverse 令 U = P ΣS, 由 U U = [ 3] =4 3 故 U U 的特徵值 λ = 4, 且對應的特徵向量為 s =, 故 U 的奇異值為 σ = 4, 且 S =[s] =, 同時 p = Us = σ 4 3 再由 可得 U p = [ 3]p = p =, p3 =

17 A.3 廣義反矩陣喻超凡 姚碩 林郁叢書 439 令 且 令 P =[p p p 3 ]= Σ= Σ + =[ ] 4 故 U 的 pseudoinverse 為 另解 : 因 U 行向量線性獨立, 故 U + = SΣ + P =[ ] U + =(U U) U =[ ] (b) 同理 V 的 pseudoinverse 為 V + = 另解 : 令 A = V T, 故 A 的行向量線性獨立, 則 A + =(A A) A =[ ] 因此 V + =(A T ) + =(A + ) T =

18 458 喻超凡 姚碩 林郁叢書第 C 章 Householder 矩陣 () k =, 則 H = I 為單位矩陣 (3) k =, 即 H = I uu T 令 w V, 則 w = w + w, 其中 w W w W, 又 u 為 W 中的單位向量, 即 u T u =, 故 < w, u >= u T w =, < w, u >= u T w = u T u w = w Hw = (I uu T )(w + w )=(I uu T )w +(I uu T )w = w uu T w + w uu T w = w + w w = w 故 H = I uu T 為 R n 映到 W 的正交投影矩陣 若 {x, x,, x n } 為 W 的一組正規化正交基底 (orthonormal basis), 令矩陣 則 H = I uu T = XX T X =[x x x n ] (4) k =, 即 H = I uu T 令 w V, 則 w = w + w, 其中 w W w W, 又 u 為 W 中的單位向量, 即 u T u =, 故 < w, u >= u T w =, < w, u >= u T w = u T u w = w Hw = (I uu T )(w + w )=(I uu T )w +(I uu T )w = w uu T w + w uu T w = w + w w = w w

19 474 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 (4) False ; 應為 至少 故選 (c) 8. Label the following statements as being true or false. (a) If an n n matrix is not invertible, then it has an eigenvector in R n. (b) We can always define infinitely many inner products for an inner product space. (c) Let A R m n and R be its reduced row echelon form. Then the span of column of R is equal to the span of columns of A. (d) If two n n matrices have the same characteristic polynomial, then they are similar. (e) If S is a linearly independent subset such that every vector in V can be written as a linear combination of the vectors in S. Then S is a basis for V. (f) Let A be an n n matrix and Av = v for every v in R n. Then A is orthogonal. (g) Every matrix in M 5 5 (R) has an eigenvector in R 5. (h) Given any A in M m n (R). AA T is always diagonalizable. (i) Let A and B be n n matrices. Suppose that B is not invertible. Then rank (AB) < rank (A). (j) Let {v, v, v 3, v 4 } be a linearly independent subset of R 5 and let T : R 5 R 4 be linear. Then {T (v ),T(v ),T(v 3 ),T(v 4 )} cannot be a linearly independent subset of R 台大電機 D (a) False, 若矩陣為實數矩陣即為正確 (b) True, 只要滿足內積的三個公設即可 (c) False, 列運算只保列空間及零核空間 (d) False, 相似矩陣具有相同的特徵多項式, 但逆定不恆真 (e) True, 基底的定義 (f) True, 正交矩陣的等價定義

20 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 Label the following statements as being true or false. (a) Let R be the reduced row echelon form of A. Then there is a unique invertible matrix P such that PA = R. (b) Let {v, v, v 3, v 4 } be a linearly independent subset of R 4 and let T : R 4 P be linear. Then {T (v ),T(v ),T(v 3 ),T(v 4 )} cannot be linearly independent. (c) If λ is an eigenvalue of an orthogonal matrix, then λ =. (d) Let R and R be row echelon form of A and A respectively. Then R + R is an row echelon form of A + A. (e) Let S be a nonempty subset of R n.thens =(S ). (f) If λ is an eigenvalue of A, thenλ k is an eigenvalue of A k for any positive integer k. (g) Let A be a n n matrix such that A k = I n for some positive integer k, them A is invertible. (h) If an n n matrix has n distinct eigenvector, then it is diagonalizable. (i) The range of a linear transformation needs not be a subspace. (j) If A is an n n skew-symmetric matrix, and n is an odd integer, them det(a) =. 94 台大電機 C (a) False ; R 為唯一, 但 P 不一定唯一, 若 A 為列滿秩時, 則 P 為唯一 (b) True ; 因 dim (P )=3, 故 4 個元素的向量集合 {T (v ),T(v ),T(v 3 ),T(v 4 )} 一定不會線性獨立 (c) True ; 正交矩陣的性質 (d) True ; 兩個 reduced row echelon form 矩陣相加, 不一定為 reduced row echelon form 矩陣, 但兩個 row echelon form 矩陣相加, 一定為 row echelon form 矩陣 (e) False ; S 要為有限維的向量空間才對 (f) True ; 設 λ 為 A 的特徵值, 且對應的特徵向量為 x, 則 Ax = λx,a x = AAx = Aλx = λ x,, A k x = λ k x

21 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 487 故 x + x 不為 A 的特徵向量 (d) False ; 令 V =(,, ) W =(,, ) Z =(,, ), 但 (V,Z), 即 V 與 Z 不正交 (e) False ; A =, 特徵向量只有 c, 但 A 可逆 (f) False ; 最小的可能的維數為, 因 dim (S + T )=dim (S)+dim (T ) dim (S T ) 3 故 dim (S T ). Determine if the following statements are true or false. () Eigenvectors of a matrix that correspond to distinct eigenvalues are orthogonal. () Every linear operator on R n has real eigenvalues. 94 暨南電機系統組 解 () False ; 要為正規矩陣才可 () False ; 要為自我伴隨運算子才成立 3. State (with a brief explanation) whether the following statements are true or false. (a) The vectors (, ), (, 3), (5, ) are linearly dependent in R. (b) The vectors (,, ), (,, ), (,, ) span R 3. (c) The set, {(,, ), (,, 3)}, is a basis for the subspace of R 3 consistingofvectorsoftheform(a, b, a 3b). (d) Any set of two vectors can be used to generate a two-dimensional subspace of R 雲科大電機

22 498 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 36. Label the following statements as being true or false. (No explanation is needed. Each correct answer gets % and each wrong answer gets %) : () The span of a nonempty subset S of R n is the largest subspace that contains S. () If A and B are two n n matrices such that AB = BA =, then either A =orb =. (3) The matrix representation of a linear operator on M n n is an n n matrix. (4) The reduced row echelon form of any orthogonal matrix is an identity matrix. (5) Let R be the reduced row echelon form of the m n matrix A with rank m. Then there is a unique invertible matrix P such that PA = R. (6) Let A be an m n matrix. Then there exist an m m orthogonal matrix U, andn n orthogonal matrix V and an m n diagonal matrix D such that A = UDV. (7) All matrices A that satisfy A A I = are invertible. (8) Let S be a finite non empty subset that spans R n.lett and U be linear operators on R n such that T (v) =U(v) for every v in S. ThenT = U. (9) The unique least norm solution to Ax = b is the orthogonal projection of b onto Col A. () A set of eigenvectors corresponding to distinct eigenvalues of a matrix is orthogonal. 95 台大電機 D () False ; 應為最小子空間 () False ; 反例為 [ ] [ ] 3 A =,B= 滿足 AB = BA =, 但 A B (3) False ; 應為 n n 的矩陣 (4) True ; 只要滿秩的方陣, 對應到的最簡列梯矩陣均為單位矩陣 (5) True ; A 為列滿秩, 則 P 為唯一

23 5 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 () True ; rank [A b] =rank (A) < 4, 故 A #μ x = #μ b 有無窮解 () True ; 此為 Cramer rule s (3) False ; 因 S T = {}, 故 dim {S T } =, 故 dim (S)+dim (T )=dim (S + T ) n (4) False ; A 行獨立, 故 m n, 則 A 具有左反矩陣才對 (5) False ; 反例為 [ ] [ ] A =,B= det(a) =det(b) 但 A 與 B 不相似 (6) True ; 定理 (7) True ; P = A(A T A) A T 為正交投影矩陣, 又 P x =, 故 x R (A) (8) False ; Proj R(A) x 必唯一 (9) False ; 要 N (L) ={} 才對 () True ; Ax = λ x A T y = λ y, 故 (Ax) T =(λ x ) T, 可得 x T A T = λ x T x T A T y = λ x T y x T λ y = λ x T y 故 (λ λ )x T y =, 因 λ λ, 故 x T y =, 即 x y 正交 38. Please state TRUE or FALSE for the following statements. (a) If A and B are invertible matrices in M n n (F) andb is similar to A, then, for any integer k>, A k and B k are similar. (b) Let T : R n R n be linear transformation. If T (x )=T(x ), then x = x when nullity (T )=. (c) If a vector space V is the direct sum of W and W then W W =. (d) {} is a linear independent set. (e) {,x,x } is an orthonormal basis for P 3 (F). (f) The vectors in an eigenspace of a linear operator T are eigenvectors of T. 95 清大電機 B

24 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 A m n matrix A is full rank if rank (A) =min(m, n). Which the following are correct? (a) A full rank AA T invertible ; AA T invertible A full rank. (b) A full rank AA T invertible ; AA T invertible A full rank. (c) A full rank AA T invertible ; AA T invertible A full rank. (d) A full rank AA T invertible ; AA T invertible A full rank. (e) All AA T s eigenvalues are strictly positive. 95 台大資工 (a) False ; 當 m>n 時, A 為 full rank 時, AA T 不可逆 (b) True ; 當 AA T 可逆時, 則 A 為 full rank (c) False ; (d) False ; (e) False ; AA T 所有的特徵值為非負的值 4. Which of the following are correct? (a) If A and B are invertible, then so is AB. (b) If A is invertible, then so is A T. (c) Eigenvalues of a triangular matrix are its diagonal elements. (d) It is impossible to have real-valued matrix A such that A = I. (e) It is possible that AB BA. 95 台大資工 (a) True ; (AB) = B A (b) True ; (A T ) =(A ) T (c) True ; 性質 [ ] [ ] (d) False; 例如 A =, A = = I (e) True ; 性質

25 58 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 nullity (A T )+rank (A T )=m 因 rank (A) =rank (A T ), 若 m n 時, 則 nullity (A) nullity (A T ) (3) True ; rank (A) =dim (R(T A )) = m = dim (R m ), 若且唯若 T A 為 onto (4) False ; 還要 V = span (S) (5) True ; 因 (,, ) V, 故 V 不為 R 3 的子空間 47. An affine transformation of R is a function T : R R of the form T (x) =Ax + b, wherea is an invertible matrix and b R.Whichof the following statements are correct? () T (x) =A x A b. () Affine transformations map straight lines to straight lines. (3) There is no affine transformation that can map a straight line to a circle. (4) Affine transformations map parallel straight lines to parallel straight lines. (5) There exists an affine transformation that maps parallel straight lines to intersecting straight lines. 97 台大電機 C () True ; 因 故 y = T (x) =Ax + b x = T (y) =A (y b) =A y A b () True ; 令直線 L 為 x = x + t v, 其中 x 為直線通過的點, v 為直線平行的向量, 故 y = T (x) =Ax + b = A(x + t v)+b = y + t u 其中 y = Ax + b u = Av, 故仍為直線 (3) True (4) True ; 設 L L 為兩平行線, 且平行向量 v, 由 () 可知轉換後兩直線均平行向量 u = Ab, 故轉換後仍為平行線

26 54 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 () False : rank (A) =rank [A b] 才有解 () True : 此為定理 (3) True : 此為定理 [ ] [ ] 4 (4) False : 不一定, 如 A = B =, 則 det(a) =det(b) =4, 但 A B 不相似 (5) False : 必須要 N (T )={}, 即 T 必須為一對一 55. True or false, MUST with reason or counterexample. (a) Let C be a nonsingular matrix and Y (x) be a fundamental matrix of the linear system y (x) =Ay(x), where y(x) is a vector function of dimension n. ThenCY (x) is also a fundamental matrix. (b) The unique solution of the following initial value problem y (x) =y(x)( y(x))(3 y(x)) sin(y(x)),y() =.5 is always increasing and between and. (c) Let a, b R and a>. Then all the solutions of y (x)+ay (x)+by(x) = have the property of y(x) asx. (d) If the Wronskian of smooth functions f (x),, f n (x) vanishesatevery point of the real line, then the n function are linearly dependent. 98 台聯 D (a) False ; 必須 AC = CA 才可使 (CY ) = CY = CAY = A(CY ) (b) True ; 因 y = y = y =3 為 ODE 的奇異解, 又初始條件為 y() =.5 介在 與 之間, 故 <y(x) <, 因此使得 y (x) > ( x R), 故 y(x) 在 x R 為遞增函數 (c) False ; 必須 b> 才會成立 (d) False : 反例如下, 令 f (x) =x f (x) =x x, 則 f (x) f (x) 在 x R

27 58 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 [ u T ] P u 3 =(I UU T )u 3 = u 3 [u u 3 ] u 3 = u 3 u 3 =u 3 故 u u 3 均為 P 的特徵向量, 對應的特徵值均為 (c) True ; (A + I) 的特徵值為,, 3, 故 det(a + I) = 3, 故 (A + I) 為可逆 u T 3 6. True and False : Suppose there are three n n matrices A, B, andc. Show whether the following statements are true or false. () If AC = BC, thena = B. () If A = A, thena =... or... (3) If both A and B are non-singular matrices, then A+B is also non-singular. (4) If both A and B are non-singular matrices, then AB is also non-singular. (5) If AB is an invertible matrix, then both A and B are invertible. 98 北科大電通甲 [ ] [ ] [ ] () False ; 如 A = B = C =, 則 AC = BC, 但 A B, 若 C 為可逆矩陣, 則命題成立 [ ] Ir () False ; 取 A =, 則 A = A n r (3) False ; 如 A = I B = I, 則 A + B = 為 singular (4) True ; (AB) = B A (5) True ; det(ab) =det(a)det(b), 故 det(a) 且 det(b)

28 5 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 (b) False ; 要 rank (A) <n (c) False ; 因 RS (A) =R n, 故 rank (A) =n, 則 A 的行向量為線性獨立 7 [ ] 7x x x (d) True ; 因 T (x) =Ax = 5 = x +5x x 4 x 4x (e) True ; 設 A B 均為 n n 的方陣, 因 AB 為可逆, 故 rank (AB) =n, 因 rank (AB) rank (A) rank (AB) rank (B), 故 rank (A) =n rank (B) =n, 即 A B 均可逆 8 (f) True ; 因 = 3 (g) False ; dim {RS (A)} = rank (A) =3 (h) False ; 不一定是對稱矩陣 (i) T : u + v = < u + v, u + v >= u + < u, v > + < v, u > + v = +++ =9 故 u + v =3 [ ][ ] [ ] x x (j) True ; 因 = y y 6. Suppose all vectors are in R n with the standard inner product space. Which of the following statements is false? (a) The set of all vectors in R n orthogonal to one fixed vector is a subspace of R n. (b) If W is a subspace of R n,thenw and W have no vectors in common. (c) If {v, v, v 3 } is an orthogonal set and if c, c, c 3 are scalars, then {c v,c v,c v 3 } is an orthogonal set. (d) If a square matrix has orthonormal columns, then it also has orthonormal rows. (e) If W is a subspace, then Proj W v + v Proj W v = v. 98 中正電機

29 53 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 79. Let V be a subspace of R n with dimension k. Which of the following statements is (are) true? (A) Every linearly independent subset of V contains at least k vectors. (B) Any finite subset of V containing more than k vectors is linearly dependent. (C) n k. (D) Any finite subset of V containing less than k vectors is linearly independent. (E) None of the above. 台大電機 C (A) False ; 應為 at most k vectors (B) True ; 性質 (C) True ; dim (R n ) dim (V ) (D) False ; 不一定 故選 (B)(C) 8. Let M be an inner product space (A) Since the inner product is an integral of a product of functions, M is a function space (B) c<u, v>=< cu, cv>for all vectors u and v in M and for every scalar c (C) <T(u), v >=< u, T(v) > for all vectors u and v in M and for every linear operator T on M (D) If M is finite-dimensional, then M contains an orthonormal basis (E) None of the preceding statements are true. 台大電機 C (A) True ; 定義 (B) False ; <cu,cv>= c c <u,v>

30 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 54 (A) True ; 相似矩陣, 具有相同的行列式值 (B) True ; 實對稱矩陣的特徵值均為實數 (C) True ; rank 的定義 (D) True ; 方陣 A 的多項式的特徵向量與 A 相同 93. Given a matrix C satisfying v T Cv for vector v in R n, answer true or false to the following statements. (a) C = C T (b) C = C (c) v T CC T v (d) The eigenvalues of C are nonnegative. (e) The column of C form a basis for a nonzero subspace of R n. 台大電機 C [ ] 4 (a) False ; 例如 C =, v R, 則 v T Cv 但 C C T (b) False ; 反例如 (a) (c) True ; 因 v T CC T v = C T v [ ] [ 3 (d) False ; 例如 C =, v = v v ], 故 v T Cv = v v v + v =(v v ) 但因 det(c λi) =λ λ +4= 故 C 的特徵值為 λ =± 3 i [ ] (e) False ; 例如 C =

31 55 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 6. Mark each of the following statements True(T) or False (F). (a) If A and B are two n n non-invertible matrices, the AB is also noninvertible. (b) If a square matrix A is not invertible, then A + I is invertible, where I is the identity matrix of the same size as A. (c) Let W be a subspace of an inner product space V,andW be the orthogonal complement of W. In general, we have W W = V. (d) We can transform any linear independent set of non-zero vectors into an orthogonal set of vectors by the Gram-Schmidt process. (e) Let T be a linear transformation from a vector space V to a vector space W. Define a transformation S : v T (v)+w form V to W,wherew is a constant vector in W.ThenSis also a linear transformation from V to W. 成大電通電磁數學 (a) T ; 因 A B 不可逆, 故 rank (A) <n rank (B) <n, 故 rank (AB) <n, 則 AB 必不可逆 [ ] [ ] (b) F ; 不一定, 例如 A = 不可逆, 而 A + I = 亦不逆 (c) F ; 應該是 W W = V (d) T ; 性質 (e) F ; 因 S() =T ()+w = w

32 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 Let A be an m n matrix. Let B be a matrix obtained by permuting two columns of A. Please indicate whether the following statements are true or false. (No proof is needed.) (a) rank (A) =rank (B) (b) The column space of A is the same as the column space of B. (c) The row space of A isthesameastherowspaceofb. (d) If m>n, then the dimension of the column space of A can not be equal to m. (e) If m = n, then the determinant of A equals the determinant of B. 3 台大電機 D (a) True ; 矩陣經由列或行基本運算 rank 不變 (b) True : 行運算保行空間 (c) False : 行運算不保列空間 (d) True : dim (CS (A)) <m (e) False : det(a) = det(b) 8. Let A and B be n n matrices. Suppose that A is diagonalizable. Please indicate whether the following statements are true or false. (No proof is needed.) (a) If B = P AP for an invertible n n matrix P,thenB is diagonalizable. (b) If A and B have the same characteristic polynomials, then B is also diagonalizable. (c) If B is diagonalizable, then A + B is diagonalizable. 3 台大電機 D (a) True : 因 A 可對角化, 故存在一個非奇異矩陣 Q, 使得 Q AQ = D

33 554 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 D 為對角矩陣, 因 B = P AP = P QDQ P 則 (P Q) B(Q P ) =(P Q) B(P Q)=D (b) False : 令矩陣 則 [ ] [ ] A =,B= det(a λ) =det(b λi) =(λ ) 但矩陣 A 可對角化, 矩陣 B 不可對角化 (c) False : 令矩陣 [ ] [ ] A =,B= 矩陣 A B 均可對角化, 但 [ ] A + B = 不可對角化 9. Let A be an m n. Letx =[x,x,,x n ] T and b =[b,b,,b m ] T be n matrix and m matrix respectively. Suppose that the system of linear equation represented by Ax = b has at most one solution for any b R m. Please indicate where the following statements are true or false. (No proof is needed.) (a) The nullity of A is. (b) rank (A) =m. (c) The rows of A are linearly independent. (d) The columns of A are linearly independent. (e) Suppose that A a matrix representation of T : R n R m for some bases. Then, T is one-to-one. (f) Let R be the reduced row echelon form of A. Then the column space of A is identical to the column space of R. 3 台大電機 D

34 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 555 A 為 m n 的矩陣, 方程式 Ax = b 至多只有一個解時, rank (A) =n (a) False : 因 nullity (A) =n rank (A) =n n = (b) False : rank (A) =m 若且唯若方程式為至少一解 (c) False : A 的行獨立, 列不一定獨立 (d) True : 因 rank (A) =n (e) True : 因 nullity (A) =, 故 T 為 - (f) False : 列運算不一定保行空間. For any vector space V, (A) If V is finite-dimensional, then no infinite subset of V is linearly independent (B) If V is finite-dimensional, then V is a subspace of R n for some positive integer n (C) If V is a function space, then V must be infinite-dimensional (D) If V is infinite-dimensional, then every infinite subset of V is linearly independent (E) None of the preceding statements are true. 3 台大電機 C 選 (A) (A) True ; (B) False ; V 不一定與 R n 有關係 (C) False ; 不一定, 如 dim {P (R)} =3 (D) False ; 不一定

35 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 557 選 (E) (A) False : 不一定, 在實數係中,det(A λi) =λ +=, 則 λ 無解 (B) False : 不一定, 應改成 A 具有少於 n 個相異的特徵值 (C) False : 不一定, 特徵值與特徵向量個數無關 (D) False : 不一定 3. Which of the following is true? (A) For any m n matrix A, AA T is always invertible (B) For any m n matrix A, rank (A) =n if and only if Ax = has only one solution. (C) For any m n matrix A, rank (A) max(n, m) (D) For an m n matrix A, ranka = m if and only if Ax = b has at least a solution for all b R m (E) None of the above 3 台大電機 C 選 (B) (C) (D) (A) False ; 必須 rank (A) =m (B) True ; 因 rank (A) =n rank (A T A)=n, 故 A T Ax = A T = x =(A T A) = (C) True ; 應為 rank (A) min(n, m) max(n, m) (D) True ; 因 rank (A m n )=m, 故 dim {RS (A)} = dim {CS (A)} = m, 則所有的 b CS (A), 若且唯若方程式至少有一解

36 558 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 4. Let A and B be n n real symmetric matrices. Which of the following statements are true? () If AB =, then all eigenvalues of BA are zero. () If AB =,thenrank (A)+rank (B) n. (3) If rank (A) =rank (B), then rank (A )=rank (B ). (4) If A =,thena =. (5) If B = I, thenb = ±I. 3 台聯 D () True ; AB BA 具有相同的特徵值 () True ; 因 AB =, 即 ABx =, x R n, 由合成變換可知, R (B) N (A), 故 rank (B) nullity (A), 又由秩零定理可知 nullity (A)+rank (A) =n 故 rank (A) +rank (B) rank (A) +nullity (A) =n (3) True ; 因 A B 為實對稱矩陣, 故必可對角化, 則存在非奇異矩陣 P Q, 及一個對角矩陣 D A D B, 使得 A = PD A P,B= QD B Q 又 rank (A) =rank (D A ) rank (B) =rank (D B ), 現 rank (A) =rank (B), 則 rank (D A )=rank (D B ), 因 D A D B 為對角矩陣, 則 rank (DA )= rank (DB ), 故 rank (A )=rank (B ) (4) True ; 因 A 為實對稱矩陣, 故必可對角化, 則存在一個非奇異矩陣 P, 及一個對角矩陣 D, 使得 A = PDP, 因 A = PD D = 故 D =, 則 D =, 故 A = PDP = [ ] (5) False ; B = I, 故 B = I.

37 56 喻超凡 姚碩 林郁叢書第 D 章嚴選是非題題庫 即 AB 與 BA 為相似矩陣 [ ] [ ] (5) True ; A = B =, 因 rank (A) rank (B), 故 A B 不 為相似矩陣, 但 [ ] A = = B 則 A B 為相似矩陣 6. Let A and B be n n real matrices. Which of the following statements are true? () If all the eigenvalues of A are positive, then x T Ax > for every nonzero x R n. () If all the eigenvalues of A are positive, then det(a + A T ) >. (3) If x T Ax > for every nonzero x R n,thendet(a) >. (4) If x T Ax < for every nonzero x R n,thendet(a) <. (5) If x T Ax > for every nonzero x R n,thendet(a + A T ) >. 3 台聯 D () False : 不一定, 除了 A 為實對稱矩陣 反例為 A = [ ] 5, x T Ax = x +5x x +x 為未定型 [ ] 5 () False ; 令 A =, 則 A 的特徵值為, 但 [ ] 5 A + A T = 5 4 且 det(a + A T )= 7 < (3) True ; 因 A 不一定為實對稱矩陣, 故原命題利用否逆命題來証, 原命題為 x R, 且 x, 使得 x T Ax >, 則 det(a) >

38 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 56 故否逆命題為 det(a), 則 x, 使得 x T Ax 因實數矩陣的特徵值為複數時, 會以共軛形式出現, 且 det(a) 為 A 的所有特徵值的積, 又因 det(a), 故 A 至少存在一個特徵值 λ, 則存在一特徵向量 x, 使得 x T Ax = x T λ x = λ x, 得証 [ ] (4) False ; 令 A =, 則 x T Ax <, 但 det(a) => (5) True ; 因 x T (A + A T )x = x T Ax + x T A T x > 又 A + A T 為對稱矩陣, 則 A + A T 為正定矩陣, 則 det(a + A T ) > 7. Which of the statements below are correct? (A) For any matrix A the matrices A T A and AA T are positive definite. (B) For any invertible matrix A, the matrices A T A and AA T are positive definite. (C) If A is positive definite, the there exists a positive definite matrix B such that B = A. (D) The sum of A and B in the last statement is a positive definite matrix. (E) None of the above. 4 台大電機 C (A) False ; A T A AA T 為 semi-positive definite. (B) True ; 因 A 為可逆, 故 A 的特徵值均不為, 則 A T A AA T 的特徵值均大於, 故為正定矩陣 (C) True ; 因 B = A, 又 A 為正定, 設 A 的特徵值為 λ, 則 λ>, 故存在一個 B, 使得 B 的特值為 λ>, 因此 B 為正定矩陣, 且 B = A (D) True ; 設 A 的特徵值為 λ, 則 λ>, 則 A + B = A + A 的特徵值為 λ + λ>, 故 A + B 為正定矩陣 選 (B) (C) (D)

39 第 D 章嚴選是非題題庫喻超凡 姚碩 林郁叢書 567 (D) T ; 設 S 為 n n 的方矩, 因 S 為行獨立, 故 rank (S) =n, 則 S 為可逆 (E) T ; 定理 5. Suppose that y (x),,y n (x) aren times differentiable functions over (, ) andw (x) denotes the Wronskian of y (x),,y n (x) atx. Which of the following statements are true? (A) W (x) vanishedateveryx if y (x),,y n (x) are linearly dependent. (B) If y (x),,y n (x) are linearly independent, then there is x such that W (x). (C) If y (x),,y n (x) are linearly independent, then W (x) forallx. (D) If y (x),,y n (x) are also solutions of an nth-order linear homogeneous ordinary differential equation with constant coefficients. Then either W (x) forallx or W (x) = for all x. (E) None of the above statements are true. 4 台聯大 C (A) T ; 定理 (B) F ; x R, 使得 W (x), 則線性獨立, 逆定不恆真 ; (C) F ; (D) T ; 定理 6. For each of the following statements, if your think the statement is correct, then give a proof to prove that the statement is correct, otherwise give a counterexample to show that the statement is incorrect () n n matrix A has n distinct eigenvalues if and only if A is diagonalizable. () Assume the matrix inverse exist, (A + B ) = A(A + B) B. (3) If any three vectors v, v, v 3 in R n are linearly independent, then the vectors w = v + v, w = v + v 3, w 3 = v + v 3 are also linear independent. (4) If A and [ B ] are tow[ n n] matrices, [ (AB) ] T = A T B T. (5) A =, B =, C = are linear independent. 4 北科大自動甲