1. 1. Let T : R2
→ R3
be the linear transformation defined by the formula
T
x
y
=


2x + 2y
−x − y
0

 .
Let A denote the standard matrix for T.
2. Set A to be the matrix
A =
0.5 0
0.25 0.5
.
For each integer k ≥ 1, let
Tk : R2
→ R2
be the linear transformation defined the formula Tk(v) = Ak
v on any vector v of R2
.
R2
. For each k ≥ 1, let’s write xk and yk for the
components of the vector Ak
v so that
Ak
v =
xk
yk
and let’s define a limit, denoted limk→∞ Ak
v, by taking the limit componentwise
lim
k→∞
Ak
v =
lim
k→∞
xk
lim
k→∞
yk
.
If T : R2
→ R2
is the function defined on an arbitrary vector v by the formula
T(v) = lim
k→∞
Ak
v,
then T is a linear transformation. Is T invertible?
3. Determine whether or not the matrices A and B below are invertible. If a given matrix
is invertible, then write the inverse as a product of elementary matrices. If a given
matrix is not invertible, then explain why it isn’t invertible.

0 0 2
1 0 1




1 0 1
2
0 1 −1


R that makes the matrix
A(t) =


0 t 0
t3
+ t2
+ t + 1 1 t2
− 1
t2
+ 1 0 t − 1


1
Assignment #2
(a) Find a basis for the null space Null(A).
(b) Find a basis for the column space Col(A).
(a) For which k ≥ 1 is the linear transformation T invertible?
(b) Fix a vector v in
(a) A = 0 −1 1
(b) B = 3 1
4. Either find a real number t ∈

2. invertible or argue that no such t exists.
5. Either show that the following subsets are subspaces of Rn
, for the specified integer n,
or explain why they aren’t.
R3
consisting of all vectors
v =


x
y
z

 x, y, z ∈ R
with |x + y + z| > 0.
R4
consisting of all vectors
v =




x1
x2
x3
x4



 x1, x2, x3, x4 ∈ R
with x1 + x2 + x3 + x4 = 0.
2
(a) The subset A of
(b) The subset S of