just read the file and give works carefully !just read the file and give works carefully !just read the file and give works carefully !just read the file and give works carefully !just read the file and give works carefully !just read the file and give works carefully !just read the file and give works carefully !

just read the file and give works carefully !just read the file and give works carefully !just read the file and give works carefully !

Name:

AMAT 220: Linear Algebra

Exam 2

April, 2020

Show all work for each problem in the space provided. If you run out of room for an answer, continue on

the back of the page. You may NOT use a calculator

Question

Points

Bonus Points

1

0

0

2

0

0

3

0

0

4

0

0

5

0

0

6

0

0

7

0

0

Total:

0

0

Score

1. Express the determinant of the following matrix A as a product of binomials

a

c

A=

0

0

x b

y

z d w

p 0 q

r 0 s

where a binomial is an expression of the form α ± β for arbitrary α, β.

2. Decide whether the following matrix A is invertible by computing its determinant

2

5

−3 −2

−2 −3 2 −5

A=

1

3 −2 2

−1 −6 4

3

3. Use Cramer’s Rule to characterize the values of s such that the following linear system has solutions

2sx1 + x2 = 1

(1)

3sx1 + 6sx2 = 2

(2)

(3)

4. Compute bases of both the null space and column space of the following matrix A, i.e. bases of N ul(A) and

Col(A), respectively.

1

2

A = 3

1

2

2

1

3

1

5

5

6

4

7

6

11 6

5 10

8

6

11 9

8

9

2

5

9

9

12

5. Compute the dimension of the subspace H = Span{v1 = (1, 2, 0)T , v2 = (−1, 1, 2)T , v3 = (3, 0, 4)T } ⊂ R3

remark: I am using the notation v = (x, y, z)T to indicate the column vector

x

v = y

z

6. Let A = {a1 , a2 , a3 } and B = {b1 , b2 , b3 } be bases for a vector space V and suppose that

a1 = 4b1 − b2

(4)

a2 = −b1 + b2 + b3

(5)

a3 = b2 − 2b3

(6)

i) Find the change of coordinates/basis matrix PB←A .

ii) Use the matrix in part i) to find [x]A given that x = 3b1 + 4b2 + b3 , that is, the A-coordinates of x.

remark: Please notice that the subscript is NOT a typo. Indeed, you must find a second matrix to find the

A-coordinates of x given ONLY the change of coordinates matrix from A to B. Please look at the last part of

4.7 or the third recorded lecture on 4.7 to figure out the trick.

7. Let R2 [x] be the vector space of polynomials of degrees less than or equal to 2, that is, as a typical vector

is of the form ax2 + bx + c where a, b, c ∈ R, that is, are real numbers… Let A = {x + 1, x − 1, 2×2 } and

E = {1, x, x2 }.

i) Compute the change of basis/coordinates matrix PE←A .

ii) Use this matrix to find the A-coordinates of −1 + 2x