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Posted: March 26th, 2021

MATA22H3 page 1

1. (a) [3 points] Find all real values c such that the vectors u = [3c2 – 2, c3 – 2, c4 – 6]and v = [10, -10, 10] are parallel.

(b) [3 points] Find the orthogonal projection of a = [1, √6, 1] on u = [4, 0, -2].

(c) [3 points] Let A = 1 5 1 7 2 8 – –24 2 . Describe all vectors b = b b b1 2 3 such that

the system Ax = b is consistent.

MATA22H3 page 2

2. [9 points] Let the points (5, -1, 2), (7, 0, -1), (9, 6, 7) be vertices of a triangle in R3.

Determine the length of each of the three sides of the triangle. Also, determine the

three interior angles of the triangle (in the form of θ = arccos(x)).

MATA22H3 page 3

3. [8 points] Find the shortest distance from the point (0, 2, -3) to the line that goes

through the points (1, -1, -2), (2, -2, -2).

MATA22H3 page 4

4. [8 points] Let ABCD be a trapezoid with sides AB and CD parallel. Let M1 and

M2 be the midpoints of the nonparallel sides (AD and BC). Use vector methods

to show that the vector —-→ M1M2 = 1

2 -→ AB + –→ DC. Hint: express 2—-→ M1M2 in terms of

–→

AD, -→ AB, –→ BC, and –→ DC.

MATA22H3 page 5

5. [9 points] (a) State the Cauchy- Schwarz inequality.

(b) Give the definition of the span of vectors v1, v2, · · · , vm ∈ Rn.

(c) Give the definition of an elementary matrix.

MATA22H3 page 6

6. [8 points] Suppose c1, c2, c3, c4, c5 ∈ R5 and they satisfy the following two equations:

2c1 + c2 +

-c1 + 2c2 – c3 – 3c4 – c5 = 0

= 0

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Furthermore it is given that the vectors c1, c2, c4 are linearly independent.

If A is the matrix that has as its ith column the vector ci for i = 1, 2, 3, 4, 5, then find

the row reduced echelon form of A. Justify your answer.

MATA22H3 page 7

7. (i) [4 points] Let A be an m × n matrix and B be an n × k matrix. Prove that

(AB)T = BTAT .

(ii) [4 points] Let A be an m × n matrix, and let c be a column vector such that

Ax = c has a unique solution.

a. Prove that m ≥ n.

b. If m = n, must the system Ax = b be consistent for every choice of b?

c. Answer part (b) for the case where m > n.

MATA22H3 page 8

8. (i) [3 points] If AT = -A, then we call A a skew-symmetric matrix. Suppose A and

B are both skew-symmetric matrices of the same size and r, s ∈ R. Prove that rA+sB

is a skew-symmetric matrix.

(ii) [5 points] Suppose that A is an m×n matrix and B is an n×n invertible matrix.

Prove that the column space of A, C(A), is equal to the column space of AB, that is

C(A) = C(AB).

MATA22H3 page 9

9. [9 points] Let A = -11 2 0 3 5 2 1 -1 1 -3 0 4 -2 -4 and b = – -7 97

(a) [6 points] Use the Gauss-Jordan method to find the general solution to Ax = b.

(b) [3 points] Give the nullspace of A.

MATA22H3 page 10

10. [8 points](a) [3 points] Give the definition of a subspace W of Rn.

(b) [5 points] Suppose that W is a subspace of Rn. Prove that

W⊥ = x ∈ Rn x · w = 0 for all w ∈ W

is also a subspace of Rn.

MATA22H3 page 11

11. [8 points] Let A =

-1 1 1 0 0 1

3 1 -1 .

Find A-1 and express it as a product of elementary matrices.

MATA22H3 page 12

12. [4 points] If A-1 =

– 2 -3 1

-1 0 2

1 2 1

and B = 2(A)T , find B-1.

13. [4 points] Find all complex numbers z satisfying z2 = i.

MATA22H3 page 13

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MATA22H3 page 14

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