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Linear Combination - Applied Linear Algebra - Exam Key, Exams of Linear Algebra

Amity University - BiharLinear Algebra

This is the Exam Key of Applied Linear Algebra which includes Homogeneous Coordinates, Linear Equations, Precise Description, Linearly Independent etc. Key important points are: Linear Combination, Statements, Explanation, Inconsistent, Set Spanned, Homogeneous, Zero Vector, Identity Matrix, Undefined, Compute

Typology: Exams

2012/2013

Uploaded on 02/18/2013

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Simon Fraser University
Math 232 Midterm 1 Solutions
1. F T T T F T
2.
0 14
3 9
13 4
AB
= − −
. For BA to be defined, the number of columns of B would have to equal
the number of rows of A. This is not the case.
3.
a.
0 1 4 5
1 3 5 2
A b
−
= −
b.
1 3 5 2 1 3 5 2 1 3 5 2 1 0 7 13
0 1 4 5 0 1 4 5 0 1 4 5 0 1 4 5
3 7 7 4 0 2 8 10 0 0 0 0 0 0 0 0
A b
− − − −
− − − −
− −
∼ ∼ ∼ ∼
This means that
1 3
2 3
3
13 7
5 4
x x
x x
x
= +
= − −
∈
, or that
1
2
3
13 7
5 4
0 1
x
x x t
x
= = − + −
.
c.
4 0 1
5 7 1 4 3
7 3 7
= − +
, and so the set
0 1 4
1 , 3 , 5
3 7 7
is not linearly independent.
4.
a.
There is no pivot in column 3. This means that the matrix is not row-equivalent to the
identity matrix, and by the Invertible Matrix Theorem, it is not invertible.
b.
82 1
3 3 3
17 5 1
3 3 3
52 1
3 3 3
1 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 0
4 1 3 0 1 0 0 1 1 4 1 0 0 1 1 4 1 0
3 2 2 0 0 1 0 2 5 3 0 1 0 0 3 5 2 1
1 0 0
0 1 0
001
A I
− − −
= − − −
− − −
∼ ∼
∼
pf2

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Simon Fraser University

Math 232 Midterm 1 Solutions

1. F T T T F T

AB

. For BA to be defined, the number of columns of B would have to equal

the number of rows of A. This is not the case.

a.

A b

b.

A b

This means that

1 3

2 3

3

x x

x x

x

, or that

1

2

3

x

x x t

x

c.

, and so the set

is not linearly independent.

a. There is no pivot in column 3. This means that the matrix is not row-equivalent to the

identity matrix, and by the Invertible Matrix Theorem, it is not invertible.

b.

8 2 1

3 3 3

17 5 1

3 3 3

5 2 1

3 3 3

A I

And so

8 2 1

3 3 3

1 17 5 1

3 3 3

5 2 1

3 3 3

A

−

a. A transformation :

n m

T  → is said to be linear if

T u + v = T u + T v

T cu = cT u

for all ,

n

u v ∈

 and all scalars c ∈ .

b.

1 1 1 1 1 1 2 2 1 2 1 2

2 2 2 2 1 1 2 2 1 2 1 2

1 1

2 2

u v u v u v u v u u v v

T T

u v u v u v u v u u v v

u v

T T

u v

1 1 1 2 1 2 1

2 2 1 2 1 2 2

u cu cu cu u u u

T c T c cT

u cu cu cu u u u

Thus T satisfies the definition in (a), and so T is linear.

c. ( ) ( )

1 2

A T e T e

a. A transformation :

n m

T  → is said to be one-to-one if whenever u ≠ v

, it follows

that

T u ≠ T v

b. For sufficiency, suppose that

T x = 0

has a non-trivial solution, x = c

. As T is linear,

x = 0

is always a solution to

T x = 0

. Since c ≠ 0

and

( )

T c = T 0 = 0

, T fails to

satisfy the definition of one-to-one.

For necessity, suppose that T is not one-to-one. Then there are vectors ,

n

a b ∈

 where

a ≠ b

, but

m

T a = c ∈

 and

( )

T b = c

. As a ≠ b

, we have that a − b ≠ 0

. Now

( ) ( ) ( )

T a − b = T a − T b = c − c = 0

and so T ( x ) = 0

has a non-trivial solution.