MA122 Mock Midterm

Name:

Time Allowed: 80 minutesTotal Value: 75 marksNumber of Pages: 7

Instructions:

Cheat Sheet: One 8:5" 11" page of study notes (both sides) is allowed as a referencewhile completing the mock test. Please note, that the cheat sheet is permittedfor the mock test only!!

Non-programmable, non-graphing calculators are permitted. No other aids allowed.

Check that your test paper has no missing, blank, or illegible pages. Note that test questions appearon both sides of the paper.

Answer in the spaces provided.

Show all your work. Insu cient justication will result in a loss of marks.

1. [4 marks] Given the following matrices,

A =

"3 1 20 1 4

#B =

"1 0 1

0 1 1

#C =

"2 0 1

1 1 0

#,

determine (A 2B)T C.

2. [4 marks] A matrix A is called skew-symmetric if AT = A.Prove that if B and C are skew-symmetric matrices, then the matrix kBC is also skew-symmetricfor any scalar k.

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3. [6 marks] Determine whether the following set A is a subspace of R4:

A =

8>>>:2664a0cd

3775 j a; c; d 2 R9>>=>>;.

4. [8 marks] Let points A (6;1; 3), B (0; 0; 5), C (2; 3; 2) and D (4; 2; 0) be the vertices of paral-lelogram ABCD.

(a) Determine the cosine of the angle at vertex A.

(b) Determine the area of the parallelogram.

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5. [6 marks] Suppose that the vectors !v1 ; !v2 and !v3 2 R4 are non-zero and mutually orthogonal(i.e., they are all perpendicular to one another). Prove that they must be linearly independent.

[Hint: Consider k1!v1 + k2!v2 + k3!v3 = !0 and take the dot product with !v1 , !v2 and !v3 in turn.]

6. [4 marks] Prove the identity for any vectors u;v 2 Rn: ku+ vk2 + ku vk2 = 2 kuk2 + 2 kvk2

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7. [8 marks] Consider the plane given by 0 : 2x 4y z = 1.

(a) Find the distance between the point P (2; 1;3) and the plane 0.

(b) Express, in scalar form, an equation of a second plane that passes through the points P (2; 1;3)and Q (3; 2;2), and is perpendicular to the given plane 0.

(c) Determine whether the plane 0 is parallel or perpendicular (or neither) to the line given by

~x =

24132

35+ t24 448

35, t 2 R.

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8. [10 marks] Consider the set of vectors from R4: S =

8>>>:26643036

3775 ;26640231

3775 ;26640220

3775 ;26642121

37759>>=>>;.

(a) Verify that the vectors in S are linearly independent.

(b) Show that the vector ~v =

26641234

3775 is in the span of S by expressing it as a linear combination ofthe vectors contained in S.

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9. [8 marks] Determine which value(s) of a will result in the following linear system

(a) being inconsistent;

(b) being consistent, with an innite number of solutions;

(c) being consistent, with a unique solution.

x+ y + z = 3

3x+ 4y + 5z = 2

2x+ a2 2 z = a 18

10. [8 marks] The augmented matrix of a system of linear equation has, through elementary row opera-tions, been reduced to the following:2666664

1 0 3 0 2 10 1 1 0 1 10 0 0 1 1 00 0 0 0 0 0

3777775.

(a) Give the general solution of the system.

(b) Provide two specic solutions to the system.

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11. [5 marks] Let L be a linear mapping with the following matrix representation:

[L] =

242 4 3 11 2 1 00 6 1 4

35(a) State the domain and codomain of L.

Domain: Codomain:

(b) Determine each of the following, if possible:

(i) L (1; 2; 3; 4) (ii) L (1; 0; 1)

12. [5 marks] Consider the operator S : R3 ! R3, dened by:

S (x1; x2; x3) = (x3; x1 + x2 + x3; x2).

Prove that S is linear.

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