Assignment 1
Do all questions. Show your work and prove your answers.
1. Solve the following system of linear equations:
1 2 3
1 2 3
1 2 3
3 3 2
4 3 4
341
x x x
x x x
x x x
+ + =
+ + =
+ + =
If a solution exists, is it unique? Why or why not?
2. Consider the plane 5z= (i.e. the set
{ }
( , , ) :|: 5x y z z =). Using standard
coordinate operations in 3
», is this a vector space?
3. Let ( )nPbe the space of polynomials of degree at most n. So,
{ }
1
1 0
( ) : ... :|:
n
n i
n a x a x a a=+ + + ∈ »P
Is this a vector space?
4. Now consider the set
{ }
1
1 0 1 0 ( )... :|: ... 0
n
n n
a a nx a x a a a+ + + + + + = ⊂ P. Is this a
subspace of ( )nP?
5. a) Consider the set
[ ]
{ }
1: 0,1 :|: (0) 0U f C f= ∈ = . Is this a subspace of
[ ]
0,1C,
the space of continuous functions with domain [0,1]?
b) How about
[ ]
{ }
2: 0,1 :|: (0) 1U f C f= ∈ = ?
6. Describe each of the null space, column space, and row space of the following
matrix:
1 4 5 6
2 1 1 3
0 2 2 2
1 3 4 5
A
−
=
.
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