MAT 1341B, INTRODUCTION TO LINEAR ALGEBRA, FALL 2003

Answers to Second Midterm, November 7, 2003

Prof. P. Selinger

Question 1. Suppose A is a 10 × 8-matrix (i.e., it has 10 rows and 8 columns). Further suppose that A has rank 3. Then how many parameters are there in the general solution of the homogeneous systems of equations Ax = 0?

Answer: There are 8 variables, of which 3 pivot variables, which leaves 5 free variables, so the answer is 5.

Question 2. If B = [ \begin{matrix}1 & 2 & 0 \\ −1 & −1 & −1 \\ 0 & 2 & −1 \end{matrix} ], then what is the diagonal of B^{−1}?

Answer: We invert B:

[ \begin{matrix}1 & 2 & 0 & 1 & 0 & 0 \\ −1 & −1 & −1 & 0 & 1 & 0 \\ 0 & 2 & −1 & 0 & 0 & 1 \end{matrix} ] ⟺ [ \begin{matrix}1 & 2 & 0 & 1 & 0 & 0 \\ 0 & 1 & −1 & 1 & 1 & 0 \\ 0 & 2 & −1 & 0 & 0 & 1 \end{matrix} ]

⟺ [ \begin{matrix}1 & 2 & 0 & 1 & 0 & 0 \\ 0 & 1 & −1 & 1 & 1 & 0 \\ 0 & 0 & 1 & −2 & −2 & 1 \end{matrix} ] ⟺ [ \begin{matrix}1 & 2 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & −1 & −1 & 1 \\ 0 & 0 & 1 & −2 & −2 & 1 \end{matrix} ]

⟺ [ \begin{matrix}1 & 0 & 0 & 3 & 2 & −2 \\ 0 & 1 & 0 & −1 & −1 & 1 \\ 0 & 0 & 1 & −2 & −2 & 1 \end{matrix} ]

So the diagonal of B^{−1} is (3, −1, 1).

Question 3. Recall that two matrices are row equivalent if each can be obtained from the other by row operations. Which, if any, of the following three matrices are row equivalent?

A = [ \begin{matrix}1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{matrix} ] ,  B = [ \begin{matrix}1 & 0 & 2 & 1 \\ 1 & 1 & 3 & 3 \\ 2 & 1 & 5 & 4 \end{matrix} ] ,  C = [ \begin{matrix}1 & −1 & 1 & −1 \\ 0 & 1 & 1 & 2 \\ 1 & 1 & 2 & 2 \end{matrix} ] ,

Answer: We reduce each of the matrices to row-canonical form. A is already in row-canonical form.

B = [ \begin{matrix}1 & 0 & 2 & 1 \\ 1 & 1 & 3 & 3 \\ 2 & 1 & 5 & 4 \end{matrix} ] ∼ [ \begin{matrix}1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 1 & 1 & 2 \end{matrix} ] ∼ [ \begin{matrix}1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{matrix} ]

C = [ \begin{matrix}1 & −1 & 1 & −1 \\ 0 & 1 & 1 & 2 \\ 1 & 1 & 2 & 2 \end{matrix} ] ∼ [ \begin{matrix}1 & −1 & 1 & −1 \\ 0 & 1 & 1 & 2 \\ 0 & 2 & 1 & 3 \end{matrix} ] ∼ [ \begin{matrix}1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & −1 & −1 \end{matrix} ]

So A and B are row equivalent to each other, but not to C.

Question 4. Which of the following statements are true in a vector space V of dimension n?

I. Any n + 1 vectors in V are linearly dependent. II. Any n + 1 vectors in V form a spanning set of V. III. Any n − 1 vectors in V are linearly independent. IV. Any n vectors in V form a basis. V. Any basis of V consists of n vectors. Answer: I and V are true. II is false; for instance, take 4 vectors in ℝ^{3} which lie in a plane. III is false; for instance, take 2 vectors in ℝ^{3} which are collinear. IV is false; for instance, take 3 vectors in ℝ^{3} which lie in a plane. Question 5. For each of the following three linear functions f, g, h : ℝ^{2} → ℝ^{2}, give its matrix representation, with respect to the standard basis on ℝ^{2}.

(a) f is a rotation by 90 degrees, as shown in the illustration:

Answer: A = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

(b) g is a reflection about the y-axis:

Answer: B = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ).

(c) h is the shearing shown in this illustration:

Answer: C = ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ).

Question 6. Let W be the subspace of ℝ^{5} spanned by the vectors u_{1} = (1, 2, −1, 3, 4), u_{2} = (2, 4, −2, 6, 8), u_{3} = (1, 3, 2, 2, 6), u_{4} = (1, 4, 5, 1, 8), u_{5} = (2, 7, 3, 3, 9).

(a) Find a subset of the vectors u_{1}, …, u_{5} which form a basis of W.

Answer: We use the casting-out algorithm. This means, we write the vectors u_{1}, …, u_{5} as the columns of a matrix and row-reduce.

[ \begin{matrix}1 & 2 & 1 & 1 & 2 \\ 2 & 4 & 3 & 4 & 7 \\ −1 & −2 & 2 & 5 & 3 \\ 3 & 6 & 2 & 1 & 3 \\ 4 & 8 & 6 & 8 & 9 \end{matrix} ] ∼ [ \begin{matrix}1 & 2 & 1 & 1 & 2 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 3 & 6 & 5 \\ 0 & 0 & −1 & −2 & −3 \\ 0 & 0 & 2 & 4 & 1 \end{matrix} ] ∼ [ \begin{matrix}1 & 2 & 1 & 1 & 2 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & −4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & −5 \end{matrix} ] ∼ [ \begin{matrix}1 & 2 & 1 & 1 & 2 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & −4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ]

The pivot columns are columns 1, 3, and 5. It follows that vectors u_{1}, u_{3}, and u_{5} form a basis of W.

(b) What is the dimension of W?

Answer: It is 3.

(c) Find a basis of ℝ^{5} which contains the vectors u_{1} and u_{3}.

Answer: We know that u_{1} and u_{3} are linearly independent. We need to add 3 linearly independent vectors to obtain a basis. Row-reduce:

[ \begin{matrix}1 & 2 & −1 & 3 & 4 \\ 1 & 3 & 2 & 2 & 6 \end{matrix} ] ∼ [ \begin{matrix}1 & 2 & −1 & 3 & 4 \\ 0 & 1 & 3 & −1 & 2 \end{matrix} ]

Since the pivot entries are in columns 1 and 2, we see that the vectors (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), and (0, 0, 0, 0, 1) are linearly independent of u_{1} and u_{3}, thus the five vectors form a basis.

Question 7. Consider the following basis of ℝ^{3}:

S = {(0, 0, 1), (0, 1, 2), (1, 2, 3)}.

Let F : ℝ^{3} → ℝ^{3} be the function defined by

F (x, y, z) = (x − y, x − z, x + y − 2z).

(a) Let v = (1, 1, 1). What are the coordinates of v with respect to the basis S?

Answer: The coordinates are [v]_{S} = [a, b, c]^{T}, where (1, 1, 1) = a(0, 0, 1) + b(0, 1, 2) + c(1, 2, 3). Solving the system of equations, we find a = 0, b = −1, c = 1, so [v]_{S} = [0, −1, 1]^{T}.

(b) Let w = (x, y, z). What are the coordinates of w with respect to the basis S?

Answer: Let

P = ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 3 \end{matrix} ) .

Then P is the change-of-coordinates matrix from S to the standard basis. P^{−1} is the change-of-coordinates matrix from the standard basis to S. We calculate

P^{−1} = ( \begin{matrix}1 & −2 & 1 \\ −2 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ) .

Then [w]_{S} = P^{−1} · [x, y, z]^{T} = (x − 2y + z, −2x + y, x).

(c) What is the matrix of F with respect to the standard basis T = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}?

Answer: The matrix of F w.r.t. the standard basis is

A = ( \begin{matrix}1 & −1 & 0 \\ 1 & 0 & −1 \\ 1 & 1 & −2 \end{matrix} ) .

(d) What is the matrix of F with respect to the basis S?

Answer: We apply F to the three basis vectors: F (0, 0, 1) = (0, −1, −2), F (0, 1, 2) = (−1, −2, −3), F (1, 2, 3) = (−1, −2, −3). We express the results in coordinates relative to S: [(0, −1, −2)]_{S} = [0, −1, 0]^{T}, [(−1, −2, −3)]_{S} = [0, 0, −1]^{T}. Therefore the desired matrix is

B = ( \begin{matrix}0 & 0 & 0 \\ −1 & 0 & 0 \\ 0 & −1 & −1 \end{matrix} )

One may also calculate it as B = P^{−1}AP.

Question 8. (a) Let F : V → W be a function between two vector spaces. Write down the axioms that F must satisfy in order to be linear.

Answer: 1. for all v, u ∈ V, F (v + u) = F (v) + F (u). 2. for all v ∈ V, k ∈ K, F (kv) = kF (v).

(b) Prove that the function F : ℝ^{3} → ℝ^{2} which is given by F (x, y, z) = (x, y + 2z) is linear.

Answer: 1. Let v = (x, y, z), u = (x′, y′, z′). Then F (v + u) = F (x + x′, y + y′, z + z′) = (x + x′, y + y′ + 2(z + z′)) = (x, y + 2z) + (x′, y′ + 2z′) = F (x, y, z) + F (x′, y′, z′) = F (v) + F (u).

2. Let v = (x, y, z) and k a scalar, Then F (kv) = F (kx, ky, kz) = (kx, ky+2kz) = k(x, y+2z) = kF (x, y, z) = kF (v).

(b) Prove that the function F : ℝ^{3} → ℝ^{2} which is given by F (x, y, z) = (x, y + 2) is not linear, by giving a concrete counterexamples to one of the axioms from (a).

Answer: For example, it fails to satisfy axiom 1 when v = (0, 0, 0) and u = (0, 0, 0). In this case we have F (v + u) = F (0, 0, 0) = (0, 2), whereas F (v) + F (u) = F (0, 0, 0) + F (0, 0, 0) = (0, 2) + (0, 2) = (0, 4).