Let L be the linear operators on ℝ 3 defined by L ( x ) = [ 2 x 1 − x 2 − x 3 2 x 2 − x 1 − x 3 2 x 3 − x 1 − x 2 ] Determine the standard matrix representation A of L , and use A to find L ( x ) for each of the following vectors x : (a) x = ( 1 , 1 , 1 ) T (b) x = ( 2 , 1 , 1 ) T (c) x = ( − 5 , 3 , 2 ) T
Let L be the linear operators on ℝ 3 defined by L ( x ) = [ 2 x 1 − x 2 − x 3 2 x 2 − x 1 − x 3 2 x 3 − x 1 − x 2 ] Determine the standard matrix representation A of L , and use A to find L ( x ) for each of the following vectors x : (a) x = ( 1 , 1 , 1 ) T (b) x = ( 2 , 1 , 1 ) T (c) x = ( − 5 , 3 , 2 ) T
Solution Summary: The author explains that L is the linear operator on R3 defined by L(x)= (0-0-0,2cdot 0-1-0
L
(
x
)
=
[
2
x
1
−
x
2
−
x
3
2
x
2
−
x
1
−
x
3
2
x
3
−
x
1
−
x
2
]
Determine the standard matrix representation A of L, and use A to find
L
(
x
)
for each of the following vectorsx:
(a)
x
=
(
1
,
1
,
1
)
T
(b)
x
=
(
2
,
1
,
1
)
T
(c)
x
=
(
−
5
,
3
,
2
)
T
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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