Review Exam
1.
Let L : V −→ W be a linear map. Show that Range(L) is a subspace
of W and Ker(L) is a subspace of V .
1
2.
 
3
Let x = 2. Find the coefficients a, b, c ∈ R, where
1
 
 
 
1
2
1





1
1
x=a
+b
+ c 1 .
0
0
1
Inother

 the coordinates of x with respect to the basis
  words
  find
2
1 
 1
1 , 1 , 1 .


0
0
1
2
3.
Let p(x) = 1 + 3x + 4x2 + 5x3 . Find the coefficients a, b, c, d ∈ R where
p(x) = a + b(x − 1) + c(x − 1)2 + d(x − 1)3 .
In other words find the coordinates
of p ∈ P4 with respect to the basis
1, x − 1, (x − 1)2 , (x − 1)3 .
3
4.
Let A and B be two similar matrices. How are det(A) and det(B)
related to each other.
4
5.
Let

1
A = 4
7
2
5
8

3
6
9
represent a linear transformation relative to the standard basis of R3 .
Furthermore put,

1
B = 2
2
−1 
1
1
5 A 2
6
2
1
4
3
1
4
3

1
5 .
6
Then B represents the same linear transformation relative to some other
basis. Write down those basis elements.
5
6.
Are the functions f (x) = cos(2x), g(x) = cos2 (x) and h(x) = 1 linearly
independent functions in C ∞ (−∞, ∞). Explain.
6
7.
What is the highest possible number of elements in a linearly independent set of vectors in P4even = {p ∈ P4 : p(−x) = p(x)}.
7
8.
Find a basis of the column space of

1 2
0 1

A=
1 0
2 4
8
the matrix

−1 0
−1 1 
.
1 −2
0 −2
9.
If A has reduced row echelon

1
0

0
0
form,
0
1
0
0
2
0
0
0
3
1
0
0

0
0
,
1
0
a) What is the dimension of the null space of A?
b) Describe the null space of A as a set.
c) Find a basis for the null space of A.
9
10.
Let


x1
 x2   
A =  .  x1
 .. 
x2
···
xn
xn
be an n × n matrix. What is the nullspace of A in terms of the vector
T
x = x1 x2 · · · xn ? Explain your answer geometrically.
10
11.
 
3
What is the orthogonal projection of the vector v = 2 onto the
1
 
3
vector w =  5 , and onto the plane P = x ∈ R3 : w · x = 0 , which
−1
is orthogonal to the vector w.
11