University at Albany Linear and Cramers Rule Algebra Worksheet

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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
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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

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