369 HW4 additional problems
1. Consider the vector space
       
0 
1
1
0



       
1
1
0
, , , 0 
V = span 








0 
0
2
2





−6
0
−3
0
Give a basis B for V . Prove that this is a basis; that is, show that the vectors in B are linearly independent
and span V . What is the dimension of V ?
2. Suppose that L : R5 → R4 is a linear map
row echelon form is

1
0

0
0
such that L(x) = Ax, where A is a matrix whose reduced
0
1
0
0
2
−2
0
0

0 8
0 5
.
1 0
0 0
Give a basis for Ker L. What is the dimension of Im L? Do we have enough information to get a basis
for Im L?
3. Suppose that L : R5 → R4 is the linear map
 


x1
x1 + 2x3 + 8x5
x2 
  x2 − 2x3 + 5x5 
 
.
L
x3  = 

x1 + x4
x4 
x2
x5
Find a matrix A such that L(x) = Ax. Then give a basis for Ker L and Im L.
3. Consider the vector space in Problem 1 above and let
 
2
2

v=
4 .
8
Show that v ∈ V . What are the coordinates [v]B of v with respect to your basis B?
4. Let V be the space of polynomials of degree at most 2, and W be the space of polynomials of degree
at most 3. Let L : V → W be the integration map: L(f ) is the indefinite integral of f . Show that L is
linear and compute the integral of 2 − x + 3x2 with matrices. Use the basis D = {1, x, x2 } for V and
R = {1, x, x2 , x3 } for W .
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