Linear algebra
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Math_270Final_Review_II.docMath 270 Final Review II
1. Is
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
+
+
3
1
1
2
1
1
2
0
1
1
0
0
1
a
a
a
a
a
invertible? Explain.2. The coordinate transformation from
2
P
to2
R
is defined as T(a+bx)=ú
û
ù
ê
ë
é
b
a
.3. Use row operations to prove that
ú
ú
ú
û
ù
ê
ê
ê
ë
é
2
2
2
1
1
1
c
c
b
b
a
a
=(a-b)(a-c)(c-b).4. Find a basis for, and state the dimension of
ï
þ
ï
ý
ü
ï
î
ï
í
ì
-
-
-
real
are
b
a
b
a
b
a
b
a
,
2
3
7
4
4
2
5. For each of the following subsets of
3
P
,which one is a subspace of P3? Explain.a.
}
0
|
)
(
{
2
3
=
+
+
+
=
a
d
cx
bx
ax
x
p
b.
}
0
|
)
(
{
2
3
¹
+
+
+
=
a
d
cx
bx
ax
x
p
c.
}
0
)
1
(
)
0
(
|
)
(
{
=
+
p
p
x
p
6. Suppose A is an mxn matrix, Ax=b, Ay=0, and b is not 0. Which statement(s) must be true? Explain.
a. b is in Row A
b. y is in
n
R
c. y is in Nul A
d. If z=x+y, then Az=b
e. A is invertible
7. W=
{
}
2
1
,
x
x
Span
where1
x
=ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
4
3
2
andú
ú
ú
û
ù
ê
ê
ê
ë
é
-
=
3
2
3
2
x
.a. Show that
{
}
2
1
,
x
x
is an orthogonal basis for Wb. Find the vector
y
ˆ
in W which is closest to y=ú
ú
ú
û
ù
ê
ê
ê
ë
é
6
4
2
c. Find the distance between y (from b) and
y
ˆ
.8. A is the matrix
ú
û
ù
ê
ë
é
4
1
2
5
. Diagonalize matrix A by finding a diagonal matrix D and an invertible matrix P such that A=PD1
-
P
9. A =.Find bases for ColA, NulA, and Row A. Be sure to state which is which.
10. Answers are equations or inequalities involving h and/or k. Write answers in the blanks. Show all work. Find all values of h and k for which A=
ú
û
ù
ê
ë
é
k
h
4
6
3
1
is the augmented matrix ofa. An inconsistent linear system______________________________
b. A linear system with many solutions__________________________
c. A linear system with a unique solution_________________________
11. Answer each question in words:
a) When is a set containing exactly two vectors linearly independent?
b) If A is an m ( n matrix, and rank A equals n, explain why m ( n
12. In this question, A is a 3 ( 3 matrix. If the statement implies A is invertible, circle Y. Otherwise circle N.
Y N a) Rank A = 3
Y N b) 0 is an eigenvalue of A
Y N c) Dim Col A + dim Nul A = 3
Y N d) Ax = b and Ay = b implies x = y
Y N e) A is diagonalizable.
Y N f ) The kernel of the linear transformation T(x) = Ax is Rn
Y N g) If B is any invertible 3 ( 3 matrix, then A is row equivalent to B
13. a) Pick any part of question 12 for which you circled N, and give an example of a matrix which satisfies the condition but which is not invertible: (For example, for part a, your answer would be a 3 ( 3 matrix whose rank is 3 but which is not invertible. For part b, your answer would be a 3 ( 3 matrix for which 0 is an eigenvalue, but which is not invertible. )
14. Prove: If the columns of B are linearly dependent, so are the columns of AB. Hint: AB = [ Ab1 Ab2 … Abp ] Use the definition of linear dependence.
15. Prove or give a counterexample: If A is a 2 ( 2 matrix, and Ax = 0, then either A is the zero matrix or x is the zero vector, or both. (Here, “Give a counterexample” means give an example of A and x, both non-zero, for which Ax = 0.)
