Let v 1 and v 2 be a basis for the vector space V , and suppose that T 1 : V → V and T 2 : V → V are the linear transformations satisfying T 1 ( v 1 ) = v 1 + v 2 T 1 ( v 2 ) = v 1 − v 2 T 2 ( v 1 ) = 1 2 ( v 1 + v 2 ) T 2 ( v 2 ) = 1 2 ( v 1 − v 2 ) Find ( T 1 T 2 ) v and ( T 2 T 1 ) v for an arbitrary vector in V and show that T 2 = T 1 − 1 .
Let v 1 and v 2 be a basis for the vector space V , and suppose that T 1 : V → V and T 2 : V → V are the linear transformations satisfying T 1 ( v 1 ) = v 1 + v 2 T 1 ( v 2 ) = v 1 − v 2 T 2 ( v 1 ) = 1 2 ( v 1 + v 2 ) T 2 ( v 2 ) = 1 2 ( v 1 − v 2 ) Find ( T 1 T 2 ) v and ( T 2 T 1 ) v for an arbitrary vector in V and show that T 2 = T 1 − 1 .
Solution Summary: The author explains how to find the value of (T_12)v for an arbitrary vector in V.
Let
v
1
and
v
2
be a basis for the vector space
V
, and suppose that
T
1
:
V
→
V
and
T
2
:
V
→
V
are the linear transformations satisfying
T
1
(
v
1
)
=
v
1
+
v
2
T
1
(
v
2
)
=
v
1
−
v
2
T
2
(
v
1
)
=
1
2
(
v
1
+
v
2
)
T
2
(
v
2
)
=
1
2
(
v
1
−
v
2
)
Find
(
T
1
T
2
)
v
and
(
T
2
T
1
)
v
for an arbitrary vector in
V
and show that
T
2
=
T
1
−
1
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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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