MATH1850 Midterm 2 Nov, 2013 2

1. (8 marks) Determine whether the set of functions 94,2,32 22 xxxxx in P2 are linearly independent. Do NOT use the Wronskian. NOTE: You must show all your calculations, and you MUST show each step of how the resulting matrix was set up. If you do this question only by inspection (i.e. trial and error or guessing), or do not show all your work, you will NOT receive marks.

MATH1850 Midterm 2 Nov, 2013 3

2. (6 marks) You are given that the reduced row-echelon form of

44294

64252

10273

54021

A is

00000

00000

1612210

3728401

Find the column space basis, row space basis, and nullspace basis of A. You must show all your work for obtaining the nullspace basis.

Final Answers: Column Space Basis: _______________________________________

Row Space Basis: ________________________________________________________

Nullspace Basis: _______________________________________________________

3. (6 marks; 3 marks each) Answer each question in the space provided.

a) Consider the set F(-∞,∞) of all real-valued functions that are defined on (-∞,∞). Suppose that scalar multiplication for this set is defined as usual, but that addition is defined instead by )(2)())(( xgxfxgf . Prove that the following vector space property does NOT hold: “(f+(g+h))(x)= ((f+g)+h)(x)”

b) Suppose that the set W consists of all 33 matrices of the form

bb

b

bb

0

00

0 where

b may be any scalar. Suppose that scalar multiplication and addition for this set is defined as usual. Determine whether this set is closed under scalar multiplication (i.e. If u is an object in W and k any scalar, then uk is in W) and prove your answer.

MATH1850 Midterm 2 Nov, 2013 4

4. (8 marks total) Answer each question in the space provided. Show ALL your work.

a) (2 marks) The planes 0)4(12)3(21)1(9 zyx and

8473 zyx are parallel. Answer TRUE or FALSE, and justify your answer. You will receive credit ONLY if your answer is correct AND properly justified. Answer: ____________ Justification:

b) (3 marks) Use the Wronskian to determine whether the functions x2 and

75 3 x are linearly independent. Other methods will NOT receive marks.

Hint: In general,

)()()(

)()()(

)()()(

)(

)1()1(2

)1(1

21

21

xfxfxf

xfxfxf

xfxfxf

xW

nn

nn

n

n

c) (2 marks) Under what conditions on scalars b and c are the vectors (5b, -1, 2) and (0, 3c, 7) orthogonal? If there are no restrictions, write N/A.

d) (1 mark) Find the distance between the points (1, 3) and (-5, 2).

MATH1850 Midterm 2 Nov, 2013 5

5. (12 marks total) Answer each of the following in the space provided. You do NOT have to show your work; only the final answer will be marked.

a) (3 marks) Find the parametric equations of the line through the point (3, 1, 2) and perpendicular to the plane 06)5(4)3(6 zyx .

Answer: x = ___________ y = ____________ z = ______________

b) (1 mark) Find the value(s) of x such that (-1, x, 6) has length 10. Answer: ____________________________________

c) (2 marks) Find the cross product of u = (a, 5, 1) and v = (-1, 3, 4) where a is a constant. Answer: vu = ____________________________________

d) (2 marks) Find a basis for the set of all 22 matrices of the form

03b

baa where a, b are real numbers.

Answer: _______________________________________________

e) (2 marks) Suppose A is a 412 matrix. What is the maximum rank that A

could have? What is the minimum nullity that AT could have? Answers: max rank (A) = _________, min nullity (AT) = _________.

f) (1 mark) Consider the set V consisting of all 22 lower triangular matrices.

Find the dimension of V. Answer: dim (V) = _____________________________

g) (1 mark) Let A = 0 1 10 0 10 0 0

. Find the rank of A2 .

Answer: rank(A2) = ____________________________

MATH1850 Midterm 2 Nov, 2013 6

6. (2 marks each; 10 marks total) For each of the following questions, select ALL of the correct answers by clearly shading in the appropriate boxes. Each question is worth 2 marks, but there may be anywhere from 0 to 4 correct answers. You will lose marks for each incorrect choice (i.e. selecting something that’s wrong, or missing something that’s correct, up to a maximum of 2 marks deduction per question (i.e. no negative marks )

a) In which of the following cases does w lie in the space spanned by u and v? [Hint: These can be done by inspection ]

w = 1 45 5

; 1 02 7

, v = 2 01 3

.

xw ; 12 xu , 2 xv . w = (4, 1, 5); u = (1, -2, 1), v = (3, -6, 3). w = (1, 1, 1, 1); u = (1, 0 ,1, 1), v = (0, 3, 0, 0).

b) Which of the following statements are true?

For every non-zero matrix A, we have rank(A) > 0.

If R is the row-echelon form (REF) of matrix A, then the nonzero row vectors in R are a basis for the row space of A.

If A is a 22 matrix with rank(A) = 2 then A is invertible.

If A is a square matrix, then the nullity of A is the same as the nullity of AT

c) Which of the following statements are true?

If span(S ) = R3, then the set S must contain at least 3 vectors

If V is a set of 2 vectors in R2 that spans R2, then V is linearly independent.

It is possible for a set of 8 vectors to span R9.

If S1 is a basis for a subspace W, and S2 is also a basis for W, then S1 and S2 must have the same number of vectors.

d) Consider the set of all 22 matrices of the form [1, b, 0, d]. Which of the following vector space properties hold?

If u and v are objects in V, then vu is in V. uvvu There is an object 0 in V called the zero vector for V, such that

uu00u For each u in V, there is an object u in V, called the negative of u, such

that 0uuuu )()(

e) Suppose it is known that A is a 56 matrix with rank(A) = 2. Which of the following are true?

The nullity of AT must be 4.

The rank of AT could be 6.

The number of parameters in the general solution of 0x A is 3. The row vectors of A form a basis for R6.