III.3.1 Written Homework Problems
MTH 214 - Linear Algebra
1. Suppose T : R3 → R2 is the linear map defined by
 
x
x


T y =
y+z
z
     
1
0 
 1





1 , 0 , 1 be a basis for R3 .
Let B =


0
1
1
2
3
Let C =
,
be a basis for R2 .
1
0
(a) Determine the matrix representation of T relative to the bases B, C denoted as AB,C .
 
1
(b) Suppose v = 1. Determine the coordinate vector [v]B .
1
(c) Compute the image T (v) and determine its coordinate vector [T (v)]C .
(d) Compute the matrix-vector product AB,C [v]B and verify that it agrees with your answer in part (c).
For the remaining problems, when the standard bases are used for both domain and codomain,
the matrix representation of T is called the standard matrix representation of T .
2. Suppose a linear map T : R3 → R4 is given.
If the standard matrix representation of T is given by the matrix

1 −2 1
 3 −4 5 

A=
0
1
1
−3 5 −4



1
9
3

then for the vector b = 
 3 , determine a vector x in R such that T (x) = b.
−6
3. Suppose T : R4 → R3 whose standard matrix representation is given by


1 −3 5 −5
A = 0 1 −3 5 
2 −4 4 −4
Determine the null space of T .
Spring 2017
MTH 214 - Prof. Lew
Page 1
III.3.1 Written Homework Problems
MTH 214 - Linear Algebra
4. Suppose T : P2 → P2 is the linear map defined as T (p(x)) = p(2x − 1).
(a) Determine the standard matrix representation of T .
(b) For p(x) = 3 + 2x − x2 , determine the coordinate vector [T (p(x))] using a matrix-vector product.
Spring 2017
MTH 214 - Prof. Lew
Page 2