True or False? In Exercises 53 and 54, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If T : R n → R m is a linear transformation such that T ( e 1 ) = [ a 11 , a 21 … a m 1 ] T T ( e 2 ) = [ a 12 , a 22 … a m 2 ] T ⋮ T ( e n ) = [ a 1 n , a 2 n … a m n ] T then the m × n matrix A = [ a i j ] whose columns corresponds to T ( e i ) is such that T ( v ) = A v for every v in R n is called the standard matrix for T . (b) All linear transformations T have a unique inverse T − 1 .
True or False? In Exercises 53 and 54, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If T : R n → R m is a linear transformation such that T ( e 1 ) = [ a 11 , a 21 … a m 1 ] T T ( e 2 ) = [ a 12 , a 22 … a m 2 ] T ⋮ T ( e n ) = [ a 1 n , a 2 n … a m n ] T then the m × n matrix A = [ a i j ] whose columns corresponds to T ( e i ) is such that T ( v ) = A v for every v in R n is called the standard matrix for T . (b) All linear transformations T have a unique inverse T − 1 .
Solution Summary: The author explains that mtimes n matrix A=left[a_ijright] is called the standard matrix for T.
True or False? In Exercises 53 and 54, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.
(a) If
T
:
R
n
→
R
m
is a linear transformation such that
T
(
e
1
)
=
[
a
11
,
a
21
…
a
m
1
]
T
T
(
e
2
)
=
[
a
12
,
a
22
…
a
m
2
]
T
⋮
T
(
e
n
)
=
[
a
1
n
,
a
2
n
…
a
m
n
]
T
then the
m
×
n
matrix
A
=
[
a
i
j
]
whose columns corresponds to
T
(
e
i
)
is such that
T
(
v
)
=
A
v
for every
v
in
R
n
is called the standard matrix for
T
.
(b) All linear transformations
T
have a unique inverse
T
−
1
.
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Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY