MAT 242: Elementary Linear Algebra
Test 2 Review
1. Let v1 = (1, 0, 1), v2 = (−1, 1, 0) and v3 = (1, 1, 2). Show that these vectors are linearly dependent and find a non-trivial linear combination of them that results in the zero vector.
2. Find bases for the nullspace, the row space and column

1 −3 −9
1 −4
A= 2
1
3
3
space of the matrix

−5
11  .
13
3. Suppose x = (4, 3) in the basis {(1, 2), (−2, 1)}. Find the coordinates of x in the basis {(5, 5), (−5, 5)}.
4. Find matrices P , D and P −1 such that A = P DP −1 where D is a diagonal matrix.
5 −3
,
A=
2
0
5. Find a basis for V ⊥ where V is the subspace of R4 spanned by v1 = (1, 3, 2, 4) and v2 = (2, 7, 7, 3).
6. Find the least-squares solution to ther overdetermined system Ax = b. Also find the orthogonal
projection of b into Col(A).


 
1
1
3
A =  1 −1  , b =  0 
3 −1
8
MAT 242, Test 2 Review
Page 1
Answer Key
1.
−2v1 − v2 + v3 = 0
2.
Null(A) = Span((3, −2, 1, 0), (−4, −3, 0, 1))
Row(A) = Span((1, −3, −9, −5), (0, 1, 2, 3))
Col(A) = Span((1, 2, 1), (−3, 1, 3))
3.
x=
4.
P =
1 3
1 2
,
D=
0.9
1.3
2 0
0 3
,
P
−1
=
−2
3
1 −1
5.
V ⊥ = Span((7, −3, 1, 0), (−19, 5, 0, 1))
6.
x=
MAT 242, Test 2 Review
11/4
13/12

23/6
p =  5/3 
43/6

,
Page 2