Find the standard matrix representation for each of the following linear operators: (a) L is the linear operator that rotates each x in ℝ 2 by 45 ° in the clockwise direction. (b) L is the linear operator that reflects each vector x in ℝ 2 about the x 1 axis and then rotates it 90 ° in the counterclockwise direction. (c) L doubles the length of x and then rotates it 30 ° in the counterclockwise direction. (d) L reflects each vector x about the line x 2 = x 1 and then projects it onto the x 1 −axis.
Find the standard matrix representation for each of the following linear operators: (a) L is the linear operator that rotates each x in ℝ 2 by 45 ° in the clockwise direction. (b) L is the linear operator that reflects each vector x in ℝ 2 about the x 1 axis and then rotates it 90 ° in the counterclockwise direction. (c) L doubles the length of x and then rotates it 30 ° in the counterclockwise direction. (d) L reflects each vector x about the line x 2 = x 1 and then projects it onto the x 1 −axis.
Solution Summary: The author describes the standard matrix representation for the linear transformation L.
Find the standard matrix representation for each of the following linear operators:
(a) L is the linear operator that rotates each x in
ℝ
2
by
45
°
in the clockwise direction.
(b) L is the linear operator that reflects each vectorx in
ℝ
2
about the
x
1
axis and then rotates it
90
°
in the counterclockwise direction.
(c) L doubles the length of x and then rotates it
30
°
in the counterclockwise direction.
(d) L reflects each vector x about the line
x
2
=
x
1
and then projects it onto the
x
1
−axis.
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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