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- September 13, 2020
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Please answer all questions(including the bonus question) accurately giving step-by-step solutions.Thank you

MAT1341B ? Test 2 ? DGD 2, 2015

19 October, 2015.

Instructor ? Anne Broadbent.

{ Family Name:

Your multiple choice answers ? 1

2 First Name: 3

Student number: {

For the marker?s use only ? 4

5

6

[Bonus] 7

Total PLEASE READ THESE INSTRUCTIONS CAREFULLY.

1. Read each question carefully, and answer all questions in the space provided after

each question. For questions 4 to 7, you may use the backs of pages if necessary, but be

sure to indicate to the marker that you have done this.

2. Questions 1 to 3 are multiple choice. These questions are worth 1 point each and no part

marks will be given. Please record your answers in the space provided above.

3. Questions 4 ? 6 and are worth 6 points each, and part marks can be earned. The correct answers here require justi?cation written legibly and logically: you must

convince the marker that you know why your solution is correct.

4. Question 7 is a challenging bonus question and is worth 3 points. It is much more di?cult

to obtain marks in the bonus question, so spend your time accordingly. You can earn 100%

without attempting Q.7.

5. Where it is possible to check your work, do so.

6. Good luck! Bonne chance! 1 1. Which of the followings are subspaces of R3 ?

U = {(xy, x, xy) | x, y ? R}

W = {(x, y, z) ? R3 | 2x ? y = 0}

V = {(x ? y, x + y, x + 5y) | x, y ? R}

X = {(x, y, z) ? R3 | x + y ? z = 0} A. Only U and V

B. Only U and W

C. Only W and X

D. Only U , V and W

E. Only W , V and X

F. Only U , V and X 2. Suppose {u, v} is a linearly independent set in vector space V , and that w ? V is chosen

so that {u, v, w} is linearly dependent. Which of the following statements is ALWAYS true?

A. {u, w} is linearly dependent.

B. {v, w} is linearly dependent.

C. {u, v} is linearly dependent.

D. w ? span{u, v}.

E. v ? span{u, w}.

F. u ? span{v, w}. 2 3. It is known that a subspace W of R77 can be spanned by 56 vectors, and that W has a

linearly independent set with 45 vectors. Then it is always true that:

A. dim W < 45

B. dim W > 45

C. 45 ? dim W ? 56

D. 45 < dim W ? 56

E. 45 ? dim W < 56

F. None of the above is true. 3 4. Let M2,2 (R) denote the vector space of 2 by 2 matrices with real entries, and de?ne

{[

U= }

a a+b

? M2,2 (R) | a, b, c ? R .

b

c a) Either check that U is closed under addition, or express U in another form so you can simply

state a theorem that guarantees that U is a subspace.

(For parts (b) and (c) you may assume that U is a subspace of M2,2 (R).)

b) Find a basis for U , and hence ?nd dim(U ).

c) Give a basis for U di?erent from the one you gave in (b).

(Remember that you must justify your answers.) 4 5 5. Recall the vector space P2 = {a + bx + cx2 | a, b, c ? R} of polynomial functions of degree

at most 2, and de?ne

X = {p ? P2 | p(5) = 0}.

a) Show that X = span{x?5, x2 ?5x}. (Hint: recall the Factor Theorem: if p is any polynomial

and p(a) = 0 for some a ? R, then p(x) = (x ? a)q(x) for some polynomial q of degree one

less than that of p.)

b) Explain why X is a subspace of P2 without using the subspace test.

c) Find a basis for X. (You may use without proof the fact proved in class that {1, x, x2 } is

linearly independent.)

d) Find dim(X).

(Remember that you must justify your answers.) 6 7 6. State whether each of the following statements is (always) true, or is (possibly) false, in the

box after the statement.

? If you say the statement may be false, you must give an explicit example – with numbers,

matrices, or functions, as is appropriate!

? If you say the statement is always true, you must give a clear explanation.

a) If {v1 , v2 , v3 } is linearly dependent in R3 , then {v1 , v2 } is also linearly dependent in R3 . ANSWER

b) The set of polynomial functions : {1 + ax2 | a ? R} is a subspace of F(R) = {f | f : R ?

R}. ANSWER 8 6 (cont.).

{[

c) }

a b

? M2,2 (R) | a + d = 0 is a subspace of M2,2 (R).

c d ANSWER d) If v1 , v2 , v3 , and v4 are linearly independent vectors in a vector space V , then dim(V ) = 4. ANSWER 9 7. [Challenge/Bonus] Suppose U and W are two 3-dimensional subspaces of M2,2 (R). Explain carefully why

there is at least one non-zero vector in U ? W = {v ? M2,2 (R) | v ? U and v ? W }. (Hint:

Assume that U ? W = {0} and ?nd a contradiction.)

Note: You cannot choose U or W . Your explanation must work for all 3dimensional subspaces U and W of M2,2 (R). 10 11