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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
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3
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Total:
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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
