The effect of several linear mappings from
R
2
{\displaystyle {}\mathbb {R} ^{2}}
Due to
fact ,
a linear mapping
φ
:
K
n
⟶
K
m
{\displaystyle \varphi \colon K^{n}\longrightarrow K^{m}}
is determined by the images
φ
(
e
j
)
{\displaystyle {}\varphi (e_{j})}
j
=
1
,
…
,
n
{\displaystyle {}j=1,\ldots ,n}
φ
(
e
j
)
{\displaystyle {}\varphi (e_{j})}
φ
(
e
j
)
=
∑
i
=
1
m
a
i
j
e
i
,
{\displaystyle {}\varphi (e_{j})=\sum _{i=1}^{m}a_{ij}e_{i}\,,}
and therefore the linear mapping is determined by the elements
a
i
j
{\displaystyle {}a_{ij}}
m
n
{\displaystyle {}mn}
a
i
j
{\displaystyle {}a_{ij}}
1
≤
i
≤
m
{\displaystyle {}1\leq i\leq m}
1
≤
j
≤
n
{\displaystyle {}1\leq j\leq n}
determination theorem ,
this holds for linear maps in general, as soon as in both vector spaces bases are fixed.
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
n
{\displaystyle {}n}
dimensional vector space
with a
basis
v
=
v
1
,
…
,
v
n
{\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}}
W
{\displaystyle {}W}
m
{\displaystyle {}m}
w
=
w
1
,
…
,
w
m
{\displaystyle {}{\mathfrak {w}}=w_{1},\ldots ,w_{m}}
For a
linear mapping
φ
:
V
⟶
W
,
{\displaystyle \varphi \colon V\longrightarrow W,}
the
matrix
M
=
M
w
v
(
φ
)
=
(
a
i
j
)
i
j
,
{\displaystyle {}M=M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )=(a_{ij})_{ij}\,,}
where
a
i
j
{\displaystyle {}a_{ij}}
i
{\displaystyle {}i}
coordinate
of
φ
(
v
j
)
{\displaystyle {}\varphi (v_{j})}
w
{\displaystyle {}{\mathfrak {w}}}
describing matrix for
φ
{\displaystyle {}\varphi }
For a matrix
M
=
(
a
i
j
)
i
j
∈
Mat
m
×
n
(
K
)
{\displaystyle {}M=(a_{ij})_{ij}\in \operatorname {Mat} _{m\times n}(K)}
φ
w
v
(
M
)
{\displaystyle {}\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M)}
v
j
⟼
∑
i
=
1
m
a
i
j
w
i
{\displaystyle v_{j}\longmapsto \sum _{i=1}^{m}a_{ij}w_{i}}
in the sense of
fact ,
is called the
linear mapping determined by the matrix
M
{\displaystyle {}M}
.
For a linear mapping
φ
:
K
n
→
K
m
{\displaystyle {}\varphi \colon K^{n}\rightarrow K^{m}}
φ
:
V
→
V
{\displaystyle {}\varphi \colon V\rightarrow V}
endomorphism ),
one usually takes the same bases on both sides. The identity on a vector space of dimension
n
{\displaystyle {}n}
Let
K
{\displaystyle {}K}
V
{\displaystyle {}V}
n
{\displaystyle {}n}
dimensional
vector space
with a
basis
v
=
v
1
,
…
,
v
n
{\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}}
W
{\displaystyle {}W}
m
{\displaystyle {}m}
w
=
w
1
,
…
,
w
m
{\displaystyle {}{\mathfrak {w}}=w_{1},\ldots ,w_{m}}
φ
⟼
M
w
v
(
φ
)
and
M
⟼
φ
w
v
(
M
)
,
{\displaystyle \varphi \longmapsto M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ){\text{ and }}M\longmapsto \varphi _{\mathfrak {w}}^{\mathfrak {v}}(M),}
defined in
definition,
are
inverse
to each other.
Proof
This proof was not presented in the lecture.
◻
{\displaystyle \Box }
