This preview shows pages 1–2. Sign up to view the full content.
Math 2940, Prelim 2 Solutions
October 27, 2011
You are NOT allowed calculators or the text. SHOW ALL WORK!
1) (a) (10 points) Find a basis for the subspace of all vectors in
R
4
orthogonal to both
(1
,
0
,
2
,
3) and (1
,
0
,
0
,

1).
(b) (8 points) Let
W
⊆
R
4
be the subspace spanned by
{
(0
,
2
,
0
,
7)
,
(1
,
1
,
1
,
4)
,
(

1
,
1
,

1
,
3)
,
(3
,
1
,
3
,
5)
}
.
What is the dimension of
W
? Find a matrix
A
whose left null space is exactly
W
.
Solution:
(a) Row reducing the matrix
±
1 0 0

1
1 0 2
3
²
we get the row reduced echelon
form
±
1 0 0

1
0 0 1
2
²
so the null space of our original matrix has basis
{
(0
,
1
,
0
,
0)
,
(1
,
0
,

2
,
1)
}
.
The null space is necessarily the orthogonal complement of the row space.
(b) Row reducing the matrix
1 1
1 4

1 1

1 3
0 2
0 7
3 1
3 5
we get the row reduced echelon form
1 0 1 1
/
2
0 1 0 7
/
2
0 0 0
0
0 0 0
0
. A basis for the null space of our matrix is then
{
(

1
/
2
,

7
/
2
,
0
,
1)
,
(

1
,
0
,
1
,
0)
}
.
Writing these as columns gives us our matrix

1
/
2

1

7
/
2
0
0
1
1
0
.
2) (8 points) a) Find the matrix
P
that projects vectors
~v
∈
R
3
onto the plane
x
+2
y

z
= 0.
b) (8 points) Express the vector (

2
,

2
,
1) as the sum of a vector in the plane from part
(a), and a vector normal to that plane.
Solution:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.