M340L Matrices, Old! Exam 2
Name:
Show all work.
4.
5.
(15 pts.) Let V=R3 (3-dimensional Euclidean space) and let
W = f(x; y; z) 2 R3 : 2x + 3y + 8z = 0g.
Show that W is a subspace of V.
(10 pts.) Suppose L:R3 ! R2 is a linear transformation, and suppose
0 1 1
2
L@ 0 A =
1
0
2.
0 1 0
1
L@1A =
;
0
The system of equations
0
1
B
B
2
B
B
B
B
@3
1
1
1
2
3
4
3
1 1
1
2
6
3
2
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
and
0 1 0
L@0A =
1
1
0
1
C
B
B
0C
0
C
B
C
B
row-reduces
to
C
B
0C
B
A
@0
1
3
1 0 0 0
1 0 0
0 1 0
0 1 0
0 0 1
0 0 1
0 0 0 0
0 0 0 1
0
What is L@
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
14
24
1
2
1 A?
3
5
9
2
11
4
1
1
1
0
If we call the left-hand side of the rst pair of matrices A, use this row-reduction
information to nd the dimensions and bases for the subspaces R(A), N(A), and R(AT ).
(5 pts. for each subspace.)
3.
0
Do the vectors @
1 0 1 0
1
1
2
1
A
@
A
@
1 , 0 ,
1 A, and
3
2
1
0 1
1
@ 3 A span R3 ?
5
Are they linearly independent?
Can you nd a subset of this collection of vectors which forms a basis for R3 ?
(10 pts. for spanning, 10 pts. for lin indep, 5 pts. for basis.)
5.
0 1
1
@
Find the orthogonal projection of the vector x = 2 A onto
0 1
0
0
the lines through @ 0 A and the vectors (a) @
0
3
1
0 1
0
1
2
1 A, (b) @ 2 A, and (c) @
1
1
1
1
2 A
1
(5 pts. each).
(20 pts.) Find, using any method (other than psychic powers), the determinant of the
matrix
1.
1
1
C
0C
C
C
C.
0C
A
1 0
1
0
1
@0
1
2 1
1 3A
4 2
4
.
Is this matrix invertible?
2.
(15 pts.) Explain why the set of vectors
W
is
3.
not
a subspace of
= f(x; y; z ) j
x
+ y + 2z = 1g
3.
R
(25 pts.) Show that the system of equations Ax = b, where
A
0
=@
1
1
0
1
1
2 A and b =
1
0
@
1
2
2 A
1
is not consistent. Find the least squares solution to this sytem, i.e., the value of
closest to b.
4.(20
pts.) For the matrix
A
0
1
@
= 2
Ax
1
2 1 0
1 3 2A
1 1 1 1
nd bases for, and the dimensions of, the row, column, and null spaces of A.
5.
(20 pts.) Find
all
of the solutions to the equation Ax = b, where
A
.
0
1
@
= 2
1
2 2 1
4 3 3 A and b =
1 2 1 2
0
@
1
2
2A
0
A friend of yours runs up to you and says `Look I've found these three vectors v1 ; v2 ; v3
in R2 that are linearly independent!' Explain how you know, without even looking at the
vectors, that your friend is wrong (again).
5.
2