Download Test 2 Review Math 3377 (30 points)

Test 2 Review
Math 3377
(30 points)
Part 1: Definitions and short answers (3 points each) [30 points total]
1. Let A be an n n matrix. A scalar c is an eigenvalue of A if Ax=cx for some nonzero vector x.
2. An eigenvector of an n n matrix A is a non-zero vector x satisfying Ax=cx for some
scalar c.
3. Let A be an n n. The characteristic polynomial p( ) is defined by
p( )=det(
)
4. Let A be an n n matrix. The characteristic equation of A is defined by
det(
)=0
5. The eigenspace of A corresponding to eigenvalue
is the set of all solutions to
6. Two n n matrices A and B are similar if there is an invertible matrix P such that
7. An n n matrix A is diagonalizable if
P is an invertible nxn matrix.
where D is a diagonal matrix and
8. A set of vectors is B=
then
linearly independent if whenever
, i.e. all of the coefficients equal 0.
9. A set of vectors is B=
combination
linearly dependent if there exists a linear
with not all of the ’s equal to 0.
10. A set of vectors B=
there exists scalars
spans the vectors space V if for every vector
with
11. A set of vectors B=
independent and B spans V.
12. A function
x=y.
is a basis for a vector space V if B is linearly
is 1-1 (one-to-one) if whenever
and f(x)=f(y) then
13. A function
f(u)=v.
14. If v is a vector in
is onto if for every
there exists some
with
, then the norm (length) of v is defined by
15. Two vectors u and v are perpendicular (or orthogonal) if their dot product,
Part 2: Computations (10 points each) [40 points total]
1. Find the eigenvalues of a matrix
2. Find a basis for the eigenspace corresponding to a given eigenvalue
3. Suppose that matrix A =
and A is factored in the form A = PDP-1
a. Find the eigenvalues of A.
b. Find a basis for each eigenspace.
4. Find the projection of a vector onto a given line.
5. Find the determinant of a matrix, and use this to solve a system of linear equations
using Cramer’s rule.
Part 3: Proofs. (10 points each). [40 points total]
1. Proofs about 1-1 or onto functions.
2. Proofs about similar matrices.
3. Proofs about linearly independent or linearly dependent sets of vectors.
4. Proofs involving dot product.
5. Proofs about eigenvalues or eigenvectors.
6. Proofs involving the null space of a linear transformation.