For the following matrices determine a cot of basis vectors

For the following matrices, determine a cot of basis vectors for the null spaces the column spaces. A = [3 2 -1 5] A = [2 1 -3 4 0 2 0 -2 8] Let P_n denote the set of all polynomial functions of degree at most n Let a_o be a feed constant. Explain why H = [p(t) = a_0 + b_x| b R} is not necessarily a vector subspace of P_x, Are there any values of a_o for which H will be a subspace? If instead H = {p(t) = a + bt| a, b R}, (i.e., the constant term is allowed to vary over all real numbers), show that H is a vector subspace of P_n.

Solution

Ans-

A matrix, in general sense, represents a
collection of information stored or arranged
in an orderly fashion. The mathematical
concept of a matrix refers to a set of numbers,
variables or functions ordered in rows and
columns. Such a set then can be defined as a
distinct entity, the matrix, and it can be
manipulated as a whole according to some
basic mathematical rules.

A matrix with 9 elements is shown below.

[
[[
[ ]
]]
]



=
==
=












aaa


















=
==
=
253
A
131211
aaa
aaa
232221
333231













819
647
Matrix [A] has 3 rows and 3 columns. Each
element of matrix [A] can be referred to by its
row and column number. For example,

=
==
=a
23
6

















A computer monitor with 800 horizontal
pixels and 600 vertical pixels can be viewed as
a matrix of 600 rows and 800 columns.   
In order to create an image, each pixel is
filled with an appropriate colour.
ORDER OF A MATRIX
The order of a matrix is defined in terms of
its number of rows and columns.

Order of a matrix = No. of rows
×
××
×
No. of
columns

Matrix [A], therefore, is a matrix of order 3
×
××
×
3.

COLUMN MATRIX
A matrix with only one column is called a
column matrix or column vector.

ROW MATRIX

















3
6
4
















A matrix with only one row is called a row
matrix or row vector.
[
[[
[ ]
]]
]
653

SQUARE MATRIX
A matrix having the same number of rows
and columns is called a square matrix.















742
942
435

















RECTANGULAR MATRIX
A matrix having unequal number of rows and
columns is called a rectangular matrix.
















1735

13145
8292

REAL MATRIX

A matrix with all real elements is called a real
matrix
PRINCIPAL DIAGONAL and TRACE
OF A MATRIX

In a square matrix, the diagonal containing
the elements a
11
, a
22
, a
33
, a
44
, ……, a
is called
the principal or main diagonal.

The sum of all elements in the principal
diagonal is called the trace of the matrix.

The principal diagonal of the matrix















742
942
435

















nn
is indicated by the dashed box. The trace of
the matrix is 2 + 3 + 9 = 14.

UNIT MATRIX

A square matrix in which all elements of the
principal diagonal are equal to 1 while all
other elements are zero is called the unit
matrix.
















001
100
010

ZERO or NULL MATRIX

A matrix whose elements are all equal to zero
is called the null or zero matrix.















000
000
000
















DIAGONAL MATRIX
If all elements except the elements of the
principal diagonal of a square matrix are
zero, the matrix is called a diagonal matrix.















002
900
030
















RANK OF A MATRIX
The maximum number of linearly
independent rows of a matrix [A] is called
the rank of [A] and is denoted by
Rank [A].
For a system of linear equations, a unique
solution exists if the number of independent
equations is at least equal to the number of
unknowns.
In the following system of linear equations

2x - 4y + 5z
= 36 … … (1)
- 3x + 5y + 7z
= 7 … … (2)
5x + 3y - 8z = - 31 … … (3)
all three equations are linearly independent.
Therefor, if we form the augmented matrix
[A] for the system where
[
[[
[ ]
]]
]



36542
A
=
==
=












31835
7753

the rank of [A] will be 3.

Consider the following linear systems with 2
independent equations.

2x - 4y + 5z = 36 … … (1)
- 3x + 5y + 7z
= 7 … … (2)
- x + y + 12z = 43 … … (3)

In the above set, Eqn. (3) can be generated by
adding Eqn. (1) to Eqn. (2). Therefore, Eqn.
(3) is a dependent equation.   
Therefor, if we form the augmented matrix
[A] for the system where
















[
[[
[ ]
]]
]



36542
A
=
==
=













431211
7753
the rank of [A] will be 2.
















MATRIX OPERATIONS

Equality of Matrices

Two matrices are equal if all corresponding
elements are equal.

[A] = [B] if
ba
=
==
=
for all i and j

[
[[
[ ]
]]
]



=
==
=
342
A












873
159
ijij















  
[
[[
[ ]
]]
]



=
==
=
342
B












873
159

Addition and Subtraction

Two matrices can be added (subtracted) by
adding (subtracting) the corresponding
elements of the two matrices.

[
[[
[ ]
]]
] [
[[
[ ]
]]
] [
[[
[ ]
]]
] [
[[
[ ]
]]
] [
[[
[ ]
]]
]
ABBAC +
++
+=
==
=+
++
+=
==
=

bac +
++
+=
==
=
ijijij

Matrices [A], [B] and [C] must have the same
order.
[
[[
[ ]
]]
]















[
[[
[ ]
]]
]



=
==
=













A
aaa
aaa
aaa
[
[[
[ ]
]]
]



=
==
=












131211
232221

333231
B
bbb
bbb
bbb
131211
232221

333231
+
++
++
++
++
++
+
=
==
=






























C
bababa
bababa
bababa
131312121111
+
++
++
++
++
++
+
+
++
++
++
++
++
+
232322222121
333332323131

Multiplication by a scalar

If a matrix is multiplied by a scalar k, each
element of the matrix is multiplied by k.

k
[
[[
[ ]
]]
]



=
==
=
A
kakaka
kakaka
kakaka












131211
232221

333231















Matrix multiplication

Two matrices can be multiplied together
provided they are compatible with respect to
their orders. The number of columns in the
first matrix [A] must be equal to the number
of rows in the second matrix [B]. The
resulting matrix [C] will have the same
number of rows as [A] and the same number
of columns as [B].
[
[[
[ ]
]]
]
=
==
=












aaa
131211
A
aaa
232221






  
[
[[
[ ]
]]
]



=
==
=
bb
B












1211
bb
bb
2221
3231

















[
[[
[ ]
]]
]




=
==
==
==
=
[
[[
[ ]
]]
] [
[[
[ ]
]]
][
[[
[ ]
]]
]






aaa
aaa
131211
232221





















BAC
bb
=
==
=



bb
bb
1211
2221
3231



























babababababa
+
++
++
++
++
++
++
++
+
+
++
++
++
++
++
++
++
+
321322121211311321121111
C
babababababa
=
==
=
322322221221312321221121
m



bac
k
kjikij

where m is the number of columns in [A] and
=
==
=
1
also the number of rows in [B].
Example:
[
[[
[ ]
]]
]












132
A
=
==
=
475






  
[
[[
[ ]
]]
]



=
==
=
32
B












65
41























[
[[
[ ]
]]
]













614332511322
C
[
[[
[ ]
]]
]
×
××
×+
++
+×
××
×+
++
+×
××
××
××
×+
++
+×
××
×+
++
+×
××
×
×
××
×+
++
+×
××
×+
++
+×
××
××
××
×+
++
+×
××
×+
++
+×
××
×
=
==
=
644735541725












2412
C
=
==
=
6737

Try the following multiplication:

[
[[
[ ]
]]
]



412
A









=
==
=



524
231



=
==
==
==
=
[
[[
[ ]
]]
] [
[[
[ ]
]]
][
[[
[ ]
]]
]
BAC



























  
1117
1229
1339
[
[[
[ ]
]]
]



=
==
=
34
B




























15
21






















Transpose of a Matrix
The transpose
[
[[
[ ]
]]
]
A
T
of an
nm×
××
×
matrix
is
the
mn×
××
×
matrix obtained by interchanging
the rows and columns of
[
[[
[ ]
]]
]



=
==
=












aaa















[
[[
[ ]
]]
]
A
.



=
==
=
254
A
[
[[
[ ]
]]
]

131211
aaa
aaa



=
==
=
232221
333231












aaa







































[
[[
[ ]
]]
]
A
692
713



=
==
=
234
A
312111
T
aaa
aaa
322212
332313
Transpose of a sum
T
T
T
[
[[
[ ]
]]
] [
[[
[ ]
]]
]
(
((
( )
))
)
[
[[
[ ]
]]
] [
[[
[ ]
]]
]
BABA +
++
+=
==
=+
++
+

Transpose of a product
T
T
T
[
[[
[ ]
]]
][
[[
[ ]
]]
]
(
((
( )
))
)
[
[[
[ ]
]]
] [
[[
[ ]
]]
]
ABBA =
==
=

672
915

Numerical example of the product rule
[
[[
[ ]
]]
]



=
==
=
32
A












15
40
[
[[
[ ]
]]
][
[[
[ ]
]]
]
(
((
( )
))
)
BA
T
T
T
[
[[
[ ]
]]
] [
[[
[ ]
]]
]
?AB















  
[
[[
[ ]
]]
]



=
==
=
=
==
=















Symmetric Matrices
A matrix
aa =
==
=
jiij
[
[[
[ ]
]]
] [
[[
[ ]
]]
]
AA =
==
=
[
[[
[ ]
]]
]
A












1034
B
=
==
=
3512
1649
22814
81211
52015




















is said to be symmetric if
for all i and j.
T







  
Example:

[
[[
[ ]
]]
]



=
==
=
234
A












072
753
















DETERMINANT OF A MATRIX

Why determinants?
In some forms of solutions for systems of
linear equations, determinants appear as
denominators in a routine manner.

In a system with 3 unknowns, the
determinant may appear in the solution in the
following way.
x
D
D
x
D
y
z
D
D
D
z
y
=
==
==
==
==
==
=
D =
==
=
aaa
aaa
131211
232221
aaa
333231
aaa
D =
==
=
131211
aaa
aaa
232221
333231


  
aaa
D =
==
=
131211
aaa
aaa
232221
333231
aaa
D =
==
=
131211
aaa
aaa
232221
333231
aa
aa
2322
11
3332


aa
a
1312
21
a +
++
+=
==
=

a
aa
3332
aa
aa
1312
31
(
((
( )
))
) (
((
( )
))
)
aaaaaaaaaa
=
==
=
32133312213223332211
(
((
( )
))
)
aaaaa
+
++
+

2213231231
2322
432
D
=
==
=
653
241
65
24
=
==
=
43
1


+
++
+
3
65
2

24
43
(
((
( )
))
) (
((
( )
))
) (
((
( )
))
)
76303868
442335463152642
=
==
=+
++
+=
==
=
×
××
××
××
×+
++
+×
××
××
××
××
××
×+
++
+×
××
×=
==
=

Find the determinant:
=
==
=D
126
413
507

  

Important Properties of Determinants

1. The value of a determinant is not altered if
its rows are written as columns in the same
order.
413
507
126

763
514
021
=
==
=
2. If any two rows ( or two columns) of a
determinant are interchanged, the
value of the determinant is multiplied
by –1.
413
507
126

126
507
413
=
==
=


3. A common factor of all elements of any row
( or column) can be placed before the
determinant.
183
3121
245
=
==
=

1243
×
××
×
×
××
×
×
××
×
3341
2145
123
4
331
215
=
==
=

4. If the corresponding elements of two rows (
or columns) of a determinant are
proportional, the value of the determinant
is zero.
1046
523
0
872
=
==
=

Meaning: Row 2 ( Row 1) is linearly
dependent on Row 1 ( Row 2). Therefore,
the linear system with three unknowns does
not have a unique solution.
5. The value of a determinant remains
unaltered if the elements of one row (or
column) are altered by adding to them any

constant multiple of the corresponding
elements in any other row ( or column).
413
507
126


124221623
×
××
×+
++
+×
××
×+
++
+×
××
×+
++
+
=
==
=
507
126

6. If each element of a row ( or a column) of a
determinant can be expressed as a sum of
two, the determinant can be written as the
sum of two determinants.
413
507
126
=
==
=
=
==
=

411
505
123

4141
+
++
+
+
++
+
+
++
+
5025
1233
414
502
123
+
++
+

= - 49

7. Determinant of a product of matrices
[
[[
[ ]
]]
][
[[
[ ]
]]
](
((
( )
))
) [
[[
[ ]
]]
] [
[[
[ ]
]]
]
BABA DDD =
==
=
[
[[
[ ]
]]
]



=
==
=
432
A












124
311















  
[
[[
[ ]
]]
]



=
==
=
321
B













413
564

[
[[
[ ]
]]
] [
[[
[ ]
]]
][
[[
[ ]
]]
]
BAC =
==
=
  
[
[[
[ ]
]]
]



=
==
=




























371026
C
26315
10116

[
[[
[ ]
]]
] [
[[
[ ]
]]
][
[[
[ ]
]]
](
((
( )
))
)
1505BAC =
==
==
==
= DD
[
[[
[ ]
]]
] [
[[
[ ]
]]
]
35B and43A =
==
==
==
= DD
[
[[
[ ]
]]
] [
[[
[ ]
]]
]
15053543BA =
==
=×
××
×=
==
=DD