(Math 2890) Review Problems II

Topics: 2.1-2.3, 2.8-2.9, 5.1-5.3, 6.1-6.3 and the materials discussed in class. Office hours before the midterm: Monday (Nov 2) 2-3:30 pm Wednesday (Nov 4) 11-12, 3-4 p.m and Friday (Nov 6) 10-12, 2-2:30 p.m

1. (a) What is a subspace in R^{n}?

(b) Is the set {(x, y, z)|x + y + z = 1} a subspace?

(c) Is the set {(x, y, z)|x − y − z = 0, x + y − z = 0} a subspace?

(d) What is a basis for a subspace?

(e) What is the dimension of a subspace?

(f) What is the column space of a matrix?

(g) What is the null space of a matrix?

(h) What is an eigenvalue of a matrix A?

(i) What is an eigenvector of a matrix A?

(j) What is the characteristic polynomial of a matrix A ?

(k) What is the subspace spanned by the vectors v_{1}, v_{2}, ⋯, v_{p}?

2. Find the inverses of the following matrices if they exist.

A = [\begin{matrix}7 & −2 \\ −4 & 1 \end{matrix}], B = [ \begin{matrix}1 & −1 & 1 \\ 2 & 3 & 1 \\ −1 & 0 & −1 \end{matrix} ], C = [\begin{matrix}2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 10 \end{matrix}].

3. (a) Let A be an 3 × 3 matrix. Suppose A^{3} + 2A^{2} − 3A + 4I = 0. Is A invertible? Express A^{−1} in terms of A if possible.

(b) Suppose A^{3} = 0. Is A invertible?

4. Find all values of a and b so that the subspace of ℝ^{4} spanned by

\{ [\begin{matrix}0 \\ 1 \\ 0 \\ −1 \end{matrix}] , [\begin{matrix}b \\ 1 \\ −a \\ 1 \end{matrix}] , [\begin{matrix}−2 \\ 2 \\ 0 \\ 0 \end{matrix}] \} is two-dimensional.

5. Let B = \{[\begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] , [\begin{matrix}3 \\ 2 \\ 1 \end{matrix} ] , [\begin{matrix}0 \\ 0 \\ 2 \end{matrix} ]\}. You can assume that B is a basis for

R^{3} (a) Which vector x has the coordinate vector [x]_{B} = [\begin{matrix}1 \\ −1 \\ 2 \end{matrix} ].

(b) Find the β-coordinate vector of y = [\begin{matrix}2 \\ −2 \\ 3 \end{matrix}].

6. Let

M = [\begin{matrix}1 & 1 & 3 & 0 \\ 1 & 2 & 5 & 1 \\ 1 & 3 & 7 & 2 \end{matrix}]

(a) Find bases for Col(M) and Nul(M), and then state the dimensions of these subspaces.

(b) Express the third column vector A as a linear combination of the basis of Col(M).

7. Find a basis for the subspace spanned by the following vectors \{[\begin{matrix}1 \\ 1 \\ 1 \end{matrix}] , [\begin{matrix}1 \\ 2 \\ 3 \end{matrix}] , [\begin{matrix}3 \\ 5 \\ 7 \end{matrix}] , [\begin{matrix}0 \\ 1 \\ 2 \end{matrix}]\}.

What is the dimension of the subspace?

8. Determine which sets in the following are bases for ℝ^{2} or ℝ^{3}. Justify your answer

(a) [\begin{matrix}−1 \\ 2 \end{matrix} ] , [\begin{matrix}2 \\ −4 \end{matrix} ] . (b) [\begin{matrix}−1 \\ 2 \\ 1 \end{matrix} ] , [\begin{matrix}1 \\ 1 \\ 0 \end{matrix} ] , [\begin{matrix}2 \\ 0 \\ 0 \end{matrix} ] . (c) [\begin{matrix}−1 \\ 2 \\ 1 \end{matrix} ] , [\begin{matrix}1 \\ 1 \\ 1 \end{matrix} ] .

(d) [\begin{matrix}−1 \\ 2 \end{matrix} ] , [\begin{matrix}1 \\ −1 \end{matrix} ] . (e) [\begin{matrix}−1 \\ 2 \\ 1 \end{matrix} ] , [\begin{matrix}1 \\ 1 \\ 0 \end{matrix} ] , [\begin{matrix}2 \\ 0 \\ 0 \end{matrix} ] , [\begin{matrix}2 \\ 1 \\ 3 \end{matrix} ] .

9. Let A be the matrix

A = [\begin{matrix}−3 & −4 \\ −4 & 3 \end{matrix}] .

Find a polynomial f(A) in A such that f(A) = 0. Verify your answer.

10. Let A be the matrix [\begin{matrix}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{matrix}] .

(a) Prove that det(A − λI) = −(λ − 1)^{2}(λ − 4).

(b) Find the eigenvalues and a basis of eigenvectors for A.

(c) Diagonalize the matrix A if possible.

(d) Find an expression for A^{k}. (e) Find an expression for the matrix exponential e^{A}.

11. Let B be the matrix [\begin{matrix}2 & 1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & 1 \end{matrix}] .

(a) Find the characteristic equation of B.

(b) Find the eigenvalues and a basis of eigenvectors for B.

(c) Diagonalize the matrix B if possible.

12. Find an basis for W^{⊥} for the following W. Verify your answer.

(a) W = Span\{[\begin{matrix}−1 \\ 2 \\ 1 \end{matrix} ]\}.

(b)W = Span\{[\begin{matrix}−1 \\ 2 \\ 1 \end{matrix} ] , [\begin{matrix}2 \\ −3 \\ 1 \end{matrix} ]\}.

13. (a) Let W = Span\{u_{1}, u_{2}\} where u_{1} = [\begin{matrix}−1 \\ 2 \\ −2 \end{matrix}] and u_{2} = [\begin{matrix}2 \\ 2 \\ 1 \end{matrix}]. Show

that {u_{1}, u_{2}} is an orthogonal basis for W.

(b)Find the closest point to y = [\begin{matrix}−1 \\ 5 \\ 1 \end{matrix}] in the subspace W.

(c) Find the distance between the point y and the subspace W.