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(MATH111)final95_99_sol.pdf
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OLD MATH 111 FINAL EXAMS (Answer not guaranteed to be correct)
Math 111 Final, Autumn 1995
(1) (15 points) Consider
..
5 .4 .24
..
3 .202
..
A= .
..
00 11 00 11
1) Find a basis of eigenvectors of A;
2) Use the resulting diagonalization to compute A4 . 3A3 +2A2 .

(2) (10 points) Consider polynomials
p1 = x 2 . x, p2 =2x 2 . 2x+1,p3 = x 2 . 2x.
1) Show that p1,p2,p3 is a basis of the space P2 of polynomials of degree 2;
2) What are the coordinates of 2x2 +3x. 1 with respect to the basis.

(3) (10 points) Consider the matrix
..
120 .1
..
131 1
..
..
A= 251 0 .
..
..
360 0 1535
Find the rank of A, the dimension of ColA, and the dimension of NulA.
(4) (15 points) Consider vector space W spanned by
...... ..
123 4
...... ..
02 .34
...... ..
u1 = ,u2 = ,u3 = ,.u4 =
...... ..
123 4 00 3 .4
1) Use Gram-Schmidt process to produce an orthogonal basis of W;
2) Find the distance from the vector (2,1,0,1) onto W;
3) Find the distance from the vector (2,1,0,1) onto W;

(5) (17 points) Consider the matrix
..
121 1
..
01 .12
..
A= .
..
251 4 11 2 .1
1) Find an orthogonal basis for NulA;
2) Extend the orthogonal basis of NulA to an orthogonal basis of R4 .

(6)
(15 points) 1) Find an example of 3 3invertible matrices A and B, such that rank(A+ B) = 1 (so that A+ B

is not invertible); 2) Find an example of 3 3matrix with orthogonal column vectors but nonorthogonal row vectors; 3) Find an example of 2 2 diagonalizable matrices A and B, such that A+ B is not diagonalizable;

(7)
(18 points) True or False (no reason needed) 1) If T : V W is a linear transformation, and {v1,v2,v3} span V, then {T(v1),T(v2),T(v3)} span W;


2) If T : V W is a linear transformation, and {v1,v2,v3} are linearly independent, then {T(v1),T(v2),T(v3)}are linearly independent;
3)If {v1,v2,,vn} span V, then dim V n;
4) If {v1,v2,,vn} span V, then {v1,v2,,vn,vn+1,,vn+m} also span V;
5) If n<dim V, then {v1,v2,,vn} are linearly independent;
6) If Ais a 1117 matrix, and the general solution of the Ax= 0 has 8 free variables, then rankA=9.
7) An nn matrix can have at most n eigenvalues;
8) If A.

= 0, then 0 is not an eigenvalue of A;
9) If A is diagonalizable, then AT is also diagonalizable;
10) If u is an eigenvector of A and B, then it is an eigenvector of A+ B;
11) If all eigenvalues of A are 1, then A= I;
12) If {u1,v1,w1} and {u2,v2,w2} are orthogonal sets, then {u1 + u2,v1 + v2,w1 + w2} is also an

orthogonal set; 13) If {v1,v2,,vn,vn+1,,vn+m} is orthonormal, then {v1,v2,,vn} is also orthonormal; 14) If the columns of a square matrix A are orthonormal, then the rows of A are also orthonormal; 15) If {u1,u2} is a basis of W, then projW v = projv+ projv;
u1u2
16) projW (u+ v) = projW u+ projW v;
17) If uV is orthogonal to all vectors in V, then u=0;
18) If U is an orthogonal matrix, then U2 is also o