Homework 7
Due: Wednesday, October 22
1. For this problem, you will be asked to prove that three different functions between finite-dimensional
vector spaces are linear transformations. In particular, you will need to show in each case that the
function respects both addition and scalar multiplication. In other words, for f : V → W, show that
(i) f (v1 + v2 ) = f (v1 ) + f (v2 ) for v1 , v2 ∈ V and (ii) f (λv) = λ f (v) for λ ∈ F, v ∈ V. The
notation is a bear, but when you get past that, these actually aren’t so bad....
(a) Consider the function
R3


x1
 x2 
x3
f
- R2
-
2x1 + x2
4x3
Show that this map is a linear transformation.
(b) Let B ∈ Mat n,n (F) (the set of all n × n matrices with entries in the field F) be some fixed
matrix. Consider the function
Mat n,n (F)
A
g-
Mat n,n (F)
- A·B−B·A
In other words, this function takes in the matrix A and outputs the matrix A · B − B · A.
Show that this is a linear transformation.
(c) Consider the function
P2 (R)[ z]
h
- R3


a+b
az2 + bz + c -  b − c 
2c
Show that this function is a linear transformation.
2. Given a matrix A ∈ Mat m,n (F), there is a corresponding linear transformation T : Rn → Rm given
by T ( x) = A · x:
1 3
(a) The matrix A =
can be used to define a linear transformation T : R2 → R2
2 4
x1
x1
via left multiplication. What is T (
)? (H INT: Compute A ·
.)
x2
x2
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008
(b) For x =
3
1
, compute both T ( x) and A · x. (H INT: They should be the same!)
3. Conversely, a linear transformation can be expressed by a matrix.
(a) Consider the linear transformation
R3


x1
 x2 
x3
f
- R3


x1 − x2
-  x1 + 4x3 
− x1
Write down a matrix A such that T ( x) = A · x.


1
(b) For x =  3 , compute both T ( x) and A · x. (H INT: They should be the same!)
−1
4. Recall that the nullspace of a linear transformation is the set of all vectors that get mapped to ~0.
Thinking of a linear transformation as a matrix A, x is in the nullspace if Ax = 0.
(a) Compute
the nullspace of f if f : R4 → R2 is given by f ( x) = Ax where A =
1 3 5 7
. Is f injective? (H INT: You should be able to express the nullspace as the
2 4 1 3
span of two vectors.)


1 3 5
(b) Compute the nullspace of g if g : R3 → R3 is given by g( x) = Bx where B =  2 4 1 .
1 1 1
Is g injective?
5. Recall that the matrix-vector product Ax can be written A1 x1 + A2 x2 + · · · + An xn where Ai is
just the i th column of A. Thus, the columns of A span the range of the linear transformation T given
by T ( x) = A · x.


1 2 3
(a) Consider the matrix A =  0 1 2 . Is the corresponding linear transformation
0 0 1
T ( x) = A · x surjective?
(b) Compute the nullspace of T (as in problem 4). Is T bijective?
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008