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Mathematics

MATH 1502

Blekherman

Fall

Description

Practice Problem for the Final Exam, Math 1502
1. Suppose that A is a 3 ▯ 3 matrix with eigenvalues 1, 2 and ▯5.
a) Find the determinant of A.
b) Is the matrix A invertible?
c) Is the matrix A diagonalizable?
d) Find the characteristic polynomial of A.
e) Find the eigenvalues of A .
f) Find the eigenvalues ofwhere B is any invertible matrix.
0 1 0 1 0 1 0 1
2 0 1 1
4 B 3C B 1 C B 0C B 7C
2. V is a subspace of R spanned by [email protected] ▯1cA @s 1 A @ 0A [email protected] ▯1 A . Let A
4 3 3 2
be the 4 ▯ 4 matrix with columns given by the 4 vectors above.
a) Find a basis of V , the dimension of V , the dimension of the kernel of A and the rank of A.
b) Is the matrix A invertible? Explain.
c) Find an orthogonal basis of V .
0 1
0
B 1C
d) Is the [email protected]▯3 A in V ? If not, ▯nd the closest vector in V to it.
4
e) Find a basis of the kernel of A.
f) Without performing further calculations, ▯nd determinant of A and explain your answer.
g) Find one eigenvalue and one eigenvector of A.
▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯
2 ▯1 1 1
3. T is a linear transformation▯3ith=T 0 and T ▯1 = ▯1
▯ ▯
(a) Find T0 .
1
▯ ▯
1
(b) Find a vector u such that T(u):=
0
(c) Find the domain of T.
(d) Find the range of T.
(e) Find the matrix of T.
(f) Without doing further calculations, ▯nd the rank of the matrix of T and explain your
answer.
(g) Does there exist a non-zero vector x such that T(x) = 0? Explain.
1 0 1 0 1
1 ▯1 ▯2
B C B C
4. Consider the equation Ax = b with A = and b = 3 C.
@ 3 4A @ 1 A
1 1 1
(a) Does this equation have a solution? If yes, ▯nd all solutions, if not, explain.
(b) Find the least squares solution of the above equation.
(c) Find the length and the dot product of the columns of A. Are the columns of A orthogonal?
(d) Find a pair of rows of A that are perpendicular to each other.
(e) Find an orthogonal basis of the column space of A.
(f) Find the dimension of the kernel of A.
5. Find the eigenvalues, eigenvectors and diagonalize the following matrices, if possible.
▯ ▯
(a) 3 ▯2 .
▯2 3
▯ ▯
(b) 3 1 .
▯1 1
0 3 0 0 0 1
B C
(c)B ▯1 1 0 0 C.
@ 2 ▯1 0 0 A
1 1 1 1
0 1
5 3 1
(d)@ 0 5 1 A.
1 0 2
6. Find the best ▯t line to the following sets of points:
(a) (1;2), (2;4), (▯1;0), (5;2), (3;3).
(b) (2;▯1), (0;0), (5;4), (▯1;2).
7. True or False. No partial credit.
(a) The span of the columns of a matrix A is equal to the range of the linear transformation T
given by T(x) = Ax.
(b) Any system of equations Ax = b has a least squares solution.
(c) Any subspace has an orthogonal basis.
(d) Any 4 linearly independent vectors in R form a basis of R .
2 (e) Any collection of non-zero orthogonal vectors is linearly independent.
(f) If the matrix A has more columns than rows then the system Ax = 0 always has in▯nitely
many solutions.
(g) Any invertible matrix can be diagonalized.
(h) Any diagonalizable matrix is invertible.
(i) If u is perpendicular to every vector in the basis of a subspace V , then the orthogonal
projection of u onto V is the zero vector.
2 2
(j) If the characteristic polynomial of A is (▯ ▯ 1) (▯ ▯ 2) then the determinant of A is 2.
(k) For an invertible matrix A, the eigenvectors of A are the same as eigenvectors of A.
(l) If a matrix A is not invertible then equation Ax = b has either no solutions of in▯nitely
many solutions.
(m) If a matrix A is invertible then equation Ax = b always has a un

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