Posted: March 26th, 2021

# MATA22H3 page 1 1. (a) [3 points] Find all real values c

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
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
MATA22H3 page 14

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