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# linear algebra

Mathematics

Homework

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(1)
Consider the system of equations
1 2 3 1
1 2 3 2
1 2 3 3
3 5 4
3 2 4
6 8
x x x b
x x x b
x x x b
+ =
+ =
+ =
Where
1 1
2 2
3 3
3 5 4
3 2 4 , ,
6 1 8
x b
A X x b b
x b
= = =
So,
AX b=
1 1
2 2
3 3
3 5 4
3 2 4
6 1 8
x b
x b
x b
=
(2)
Corresponding to matrix
A
there is a linear transformation
T
, what is the transformation
and this transformation goes from what space to what space
If
:T V W
be a linear transformation and
{ }
1 2
, ,...,
n
B x x x=
and
be
basis for
V
and
W
then in order to write down the matrix of
T
relative to the basis
B
and
B
So this transformation goes to one-to-one correspondence between linear transformations
:
n m
T
¡ ¡
and
m n
×
matrices.
Hence
3 3 3 1× ×
¡ ¡
(3)
What is the determinant for matrix A and what does this tell you about matrix A being
invertible or not
Determinant of a square matrix:
A square matrix having same number of rows and columns it will have
2
n n n× =
array of
numbers. These
2
n n n× =
numbers also determine a determinant having
n
rows and
n
columns and is denoted by
or Det A A
If
( )
0det A
, we have
( ) ( )
( )
det
Aadj A adj A A
A I
=
=
Hence
A
is invertible.
Hence the determinant for matrix A and what does this tell you about matrix A being
invertible or not
Zero.
(4)
What is a basis for the Null Space of
A
, what is the rank of the Null Space and what does
this tell you about the linear transformation being one-to-one.
Consider the matrix by Gaussian-Jordan elimination and the null space of
A
is the set of
X
such that
0AX =
and reduce matrix and the equation with zero on the R.H.S
Let
( )
ij
A a=
be an
m n
×
matrix , the nullspace of the matrix
A
denoted
( )
N A
and all
n
dimensional column vectors
X
such that
0AX =
So,
rank nullity+
is the number of all columns in the matrix
A
So this transformation goes to one-to-one correspondence between linear transformations
:
n m
T
¡ ¡
and
m n×
matrices.
Rank of null Space is
dimension 3
That implies it is Linear transformation
(5)
What is the dimension of the column space of
A
and what does this tell you about the
linear transformation being onto or not
Column space
( )
__ __ __
1 2
, ,...,
n
C A Span a a a
=
÷
( )
for CBasis A=
( )
span C
independent
Vector A
Linear
=
=
We have
:
n m
T
¡ ¡
be a linear transformation then
T
is onto.
(6)
Consider
2
5 3
4 4
3 3
4 4
L
=

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5 3
2
1 1
4 4
mapped to
2
1 3 3 1
3
4 4
=
5 3
1
1 1
4 4
mapped to
2
1 3 3 1
0
4 4
=
5 3
1
1 1
4 4
mapped to
2
1 3 3 1
0
4 4
=
5 3
2
1 1
4 4
mapped to
3
1 3 3 1
2
4 4
=
(7)
In the previous problem what is the area of the original square and what is the area of the
new object under the transformation?
5 5 29
1 1
5 3
3
2 2
8 8 8
2 2
4 4
2 3
3 3 3 3 21
0 0
2
3 2
4 4 8 8 8
=
(8)
What is the determinant of matrix
2
L
and is matrix
2
L
invertible. If it is invertible then
find the inverse?
Determinant of matrix
2
L
is,
5 3 3 3 15 9
4 4 4 4 16 16
6
16
3
8
=
÷ ÷ ÷ ÷
=
=
2
5 3
4 4
3 3
4 4
L
=
We have
1
d b
A
c a
ad bc
=
1
2
2 2
10
2
3
L
=
(9)
If
2
L
is invertible what does this tell you about the dimension of the Null Space of
matrix and is the corresponding linear transformation one-to-one or not?
The dimension of the Null Space of
2
5 3
4 4
3 3
4 4
L
=
is,
Since
1
2
2 2
10
2
3
L
=
So dimension of Null Space is
2
The linear transformation one-to-one
(10)
Consider the matrix
3 5
2 4
A
=
( ) ( )
( ) ( )
2
2
2
3 5
2 4
3 4 10
12 3 4 10
2
2 2
1 2
A I
λ
λ
λ
λ λ
λ λ λ
λ λ
λ λ λ
λ λ
=
= +
= + + +
= +
= +
= +
So,
( )
det 0
1, 2
A I
λ
λ
=
=
Therefore, the eigenvalues of
A
is
1, 2
λ
=
Case(1) let
1
λ
=
Eigen vectors
X
corresponding to the Eigen root
1
λ
=
given by
( )
1 0A I X
=

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1
2
3 1 5
2 4 1
2 5 0
2 5 0
x
x
=
2 2 1
R R R= +
1
2
2 5 0
0 0 0
x
x
=
1 2
1 2
2
1
2 5 0
2 5
5
2
x x
x x
x
x
+ =
=
=
Let
2
x k=
then
1
5
2
k
x
=
Therefore, Eigen vectors corresponding to Eigen root
1
λ
=
are given by
5
2
1
k
where
k
is a non-zero parameter
Clearly the subspace generated by
5
2
1
is a one-dimensional characteristic space of
2
v
Case(1) let
2
λ
= −
Eigen vectors
X
corresponding to the Eigen root
1
λ
=
given by
( )
1 0A I X =
1
2
1
2
3 2 5
2 4 2
5 5 0
2 2 0
1 1 0
1 1 0
x
x
x
x
+
+
=
=
2 2 1
R R R= +
1
2
1 1 0
0 0 0
x
x
=
1 2
1 2
0x x
x x
+ =
=
Let
2
x k=
then
1
x k= −
Therefore, Eigen vectors corresponding to Eigen root
2
λ
= −
are given by
1
1
k
where
k
is a non-zero parameter
Clearly the subspace generated by
1
1
is a one-dimensional characteristic space of
2
v
(11)
Consider the matrix
3 4
5 5
A
=
( ) ( )
3 4
5 5
3 5 20
A I
λ
λ
λ
λ λ
=
= +
So,
( )
det 0
1 2 , 1 2
A I
i i
λ
λ
=
= − +
Therefore, the eigenvalues of
A
is
1 2 , 1 2i i
λ
= − +
Case(1) let
1 2i
λ
= − +
Eigen vectors
X
corresponding to the Eigen root
1
λ
=
given by
( )
1 0A I X =
( )
( )
1
2
3 1 2 4
5 5 1 2
4 2 4 0
5 4 2 0
i
i
x
i
x
i
+
+
=
( )
( )
1 2
1 2
4 2 4 0
5 4 2 0
i x x
x i x
+ =
+ =
Therefore, Eigen vectors corresponding to Eigen root
1 2i
λ
= +
Clearly the subspace generated by
4 2
5 5
1
i
is a one-dimensional characteristic space of
2
v
Case(1) let
1 2i
λ
= −
Eigen vectors
X
corresponding to the Eigen root
1
λ
=
given by
( )
1 0A I X =

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(1)Consider the system of equations Where So, (2)Corresponding to matrix there is a linear transformation , what is the transformation and this transformation goes from what space to what spaceIf be a linear transformation and and be basis for and then in order to write down the matrix of relative to the basis and So this transformation goes to one-to-one correspondence between linear transformations and matrices.Hence (3)What is the determinant for matrix A and what does this tell you about matrix A being invertible or notDeterminant of a square matrix:A square matrix having same number of rows and columns it will have array of numbers. These numbers also determine a determinant having rows and columns and is denoted by If , we have Hence is invertible.Hence the determinant for matrix A and what does this tell you about matrix A being invertible or notZero.(4)What is a basis for the Null Space of , what is the rank of the Null Space and what does this tell you about the linear transformation being one-to-one.Consider the matrix by Gaussian-Jordan elimination and the null space of is the set of such that and reduce matrix and the equation with zero on the R.H.S Let be an matrix , the nullspace of the matrix denoted and all dimensional column vectors such that So,is the number of all columns in the matrix So this transformation goes to one-to-one correspondence between linear transformations and matrices.Rank of null Space is That imp ...
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