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Math 113 Final Exam

Dec 15, 1997

(1) (10 points) Consider polynomial

3 323
p1 =1+ t. t,p2 =2t. t2 . t,p3 =1 . t+ t,p4 =2+ t2 . t
of P3, the vector space of polynomials of degree 3. Find a basis for span{p1,p2,p3,p4}.
(2) (15 points) Consider vector space W spanned by
.. ..
11
.. ..
. 1 .. 1 .
u1 = .. ,u2 = .. .
. 0 .. 1 . 01
1) Find a basis for the orthogonal compliment of W in R4;
2) Find the distance from
..
0
..
. 0 .
..
. 0 . 12
to W.
(3) (12 points) Consider the matrix
..
1114
..
A= . 1012 . . 0102
1) Find an orthogonal basis for ColA; 2) Find a basis for NulA; 3) Find the rank of AT .
(4) (13 points) Let T be the orthogonal projection of R4 onto the span of
.. ..
11
.. ..
. 1 .. 1 .
u1 = .. ,u2 = .. .
. 0 .. 1 . 01
1
Then T is a linear transformation from R4 to R4 . 1) Find the standard matrix for T. 2) Diagonalize T.
(5) (10 points) Let V be the vector space of 2 2 matrices. (We know V is isomorphic to R4.) Let W be the subspace spanned by

1 .1 1 .1 1 .1
v1 = , v2 = , v3 = .
0 .1 1 1 2 c

For which value of c, W is
1) 3-dimensional,
2) 2-dimensional.
(6)
(15 points) 1) Let P be the orthogonal projection of Rn onto a subspace W of Rn . Then P is a linear transformation from Rn to Rn. Explain why P is diagonalizable. 2) Let A bea2 3 matrix, AT be the transpose of A. Explain why AT A can not be invertible.

3) If a square matrix P satis.es P2 . 3P +2I = O, where I is the identity matrix, and O is the zero matrix, then the eigenvalues of P must be either 1 or 2. Explain.

(7)
(12 points) 1) Find an example of 3 3 diagonalizable matrices A and B, such that A+ B is not a diagonalizable matrix; 2) Find an example of two 3 3 matrices A and B of rank 2 such that A + B is of rank 3. 3) Find an example of 3 3 matrix A such that A is not a diagonalizable matrix, but A3 is diagonalizable.

4) Find an example of 2 2 matrices A and B, such that A and AB is diagonalizable, but B is not diagonalizable.

(8)
(13 points) True or False (no reason needed) 1) If v1,v2,v3,v4 span a 4-dimensional vector space V, then {v1,v2,v3,v4} is a basis of V ; 2) An injective linear transformation maps a linear independent set of vectors to a linear independent set of vectors; 3) If an invertible n n matrix A has few than n distinct eigenvalues, then A is not


diagonalizable;
4) If v is an eigenvector of A and A is invertible, then v is an eigenvector of A.1;
5) If A is diagonalizable, then AT is also diagonalizable;
6) If u and v are eigenvectors of A correspondingto the distinct eigenvalue, then u+v is not an eigenvector of A;
7) If v is an eigenvector, then any scale multiple of v is also an eigenvector.
8) If A and B are two similar n n matrices, then A and B have the same set of
eigenvalues, moreover, the corresponding eigenspaces are same; 9) If U is orthogonal 3 3 matrix and {v1,v2,v3