May 2024
Intermediate to advanced
687 pages
18h 35m
English
Content preview from Discrete Mathematical Structures by Pearson
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12-38 Discrete Mathematical Structures
12.7.2 Matrix Representation of a Linear Transformation
Let T: R
n
→ R
m
be a linear transformation. Let
…
ee e
n12
,,,
be the standard basis in R
n
. Let us
define Le aa
aa
jj n
…
=
()
and A=
12
,,,
Now if
~
x ∈ R
n
, then we have
Tx Txex
ex
e
nn
=+++()
11 22
=+ ++ =
()
=xa xa xa aa ax Ax
nn n1
…
12212
,,,
Thus, linear transformation T can be represented by m × n matrix A.
Matrix of Linear Transformation: Let V and W be finite dimensional vector spaces over the
same field F with dimensions n and m, respectively. Also, let E
1
= {
~
v
1
,
~
v
2
,…,
~
v
n
} and E
2
= {
~
w
1
,
~
w
2
,…,
~
w
n
} be two ordered bases for V and W, respectively.
An element
vV∈
can be uniquely expressed as
vxvx
vx
v
nn
=+++
11 22
where x
1
, x
2
,…, x
n
are ...
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