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Final Exam Ma 122 A Roden Jason A. David 27 March 2009 • Write only on one side of the paper. • Show your complete solution. • Encircle your final answer. 1. For each of the following statements, if the statement is true, then write T. If the statement is false, replace the underlined phrase with the correct words to make the statement true. [10 pts. each] a. A linear system with more variables than equations has infinitely many solutions. b. Two vector spaces are isomorphic if and only if they have the same dimension. c. A matrix A G R n x n is positive definite if and only if X T J 4 X > 0 for every x in IR n d. A linear transformation L : V i—> W is one-to-one if and only if x = y implies L(x) = L(y) for every x and y in V. e. Let L : H n i-» IR m be a linear transformation defined by L(x) = A x for every x G IR n where A G IR m X n . Let r = rank (A). L is onto if and only if r = m < n . 2. Find all eigenvalues and the associated eigenvectors of the matrix [25 pts.] A 3. If (A, x) is an eigenpair of A G ]R nxri , show that (p(A), x) is an eigenpair ofp(A) where p(t) is polynomial function in t. [25 pts.] 4. Let U and W be vector subspaces of a vector space V. Show that U + W = {u + w|u eW,w£ W} is a subspace of V. [25 pts.] 5. Let A be a 3 x 3 nonsingular matrix written column-wise as A = (xi x 2 x 3 ) where Xj G 1R3 for i = 1,2, 3. Suppose A 1 = T Y2 where G H 3 for i — 1,2, 3. Vy 3 T / If B = (xi ax 2 x 3 ) where a is a nonzero scalar, write B 1 in terms of y x , y 2 , and y 3 . [25 pts.]
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David Test 1
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F i n a l E x a m
M a 122 A R o d e n J a s o n A . D a v i d 27 M a r c h 2009
• Write only on one side of the paper. • Show your complete solution. • Encircle your final answer.
1. For each of the following statements, if the statement is true, then write T . If the statement is false, replace the underlined phrase w i th the correct words to make the statement true. [10 p t s . each]
a. A linear system w i t h more variables than equations has infinitely many solutions.
b. Two vector spaces are isomorphic if and only if they have the same dimension.
c. A matr ix A G R n x n is positive definite if and only if X T J 4 X > 0 for every x in IR n
d. A linear transformation L : V i—> W is one-to-one if and only if x = y implies
L ( x ) = L ( y ) for every x and y in V.
e. Let L : H n i - » I R m be a linear transformation defined by L ( x ) = A x for every x G IR n where A G I R m X n . Let r = rank ( A ) . L is onto if and only if r = m < n .
2 . F i n d al l eigenvalues and the associated eigenvectors of the matr ix [25 pts.]
A
3. If (A, x) is an eigenpair of A G ] R n x r i , show that (p(A), x ) is an eigenpair o f p ( A ) where p ( t ) is polynomial function i n t . [25 pts.]
4. Let U and W be vector subspaces of a vector space V . Show that
U + W = {u + w|u e W , w £ W }
is a subspace of V . [25 pts.]
5. Let A be a 3 x 3 nonsingular matr ix written column-wise as
A = ( x i x 2 x 3 )
where X j G 1R 3 for i = 1 , 2 , 3. Suppose A 1 = T Y 2 where G H 3 for i — 1,2, 3.
V y 3 T / If B = ( x i a x 2 x 3 ) where a is a nonzero scalar, write B 1 in terms of y x , y 2 , and y 3 . [25 pts.]
