The cross product of two vectors in ℝ 3 is given by [ a 1 a 2 a 3 ] × [ b 1 b 2 b 3 ] = [ a 2 b 3 − a 3 b 2 a 3 b 1 − a 1 b 3 a 1 b 2 − a 2 b 1 ] . See Definition A.9 and Theorem A.11 in theAppendix. Consider an arbitrary vector v → in ℝ 3 . Is thetransformation T ( x → ) = v → × x → from ℝ 3 to ℝ 3 linear?If so. find its matrix in terms of the components of thevector v → .
The cross product of two vectors in ℝ 3 is given by [ a 1 a 2 a 3 ] × [ b 1 b 2 b 3 ] = [ a 2 b 3 − a 3 b 2 a 3 b 1 − a 1 b 3 a 1 b 2 − a 2 b 1 ] . See Definition A.9 and Theorem A.11 in theAppendix. Consider an arbitrary vector v → in ℝ 3 . Is thetransformation T ( x → ) = v → × x → from ℝ 3 to ℝ 3 linear?If so. find its matrix in terms of the components of thevector v → .
Solution Summary: The author explains that the function T is a linear transformation from Rmto Rn.
The cross product of two vectors in
ℝ
3
is given by
[
a
1
a
2
a
3
]
×
[
b
1
b
2
b
3
]
=
[
a
2
b
3
−
a
3
b
2
a
3
b
1
−
a
1
b
3
a
1
b
2
−
a
2
b
1
]
. See Definition A.9 and Theorem A.11 in theAppendix. Consider an arbitrary vector
v
→
in
ℝ
3
. Is thetransformation
T
(
x
→
)
=
v
→
×
x
→
from
ℝ
3
to
ℝ
3
linear?If so. find its matrix in terms of the components of thevector
v
→
.
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.
a. Write the vector (-4,-8, 6) as a linear combination of a₁ (1, -3, -2), a₂ = (-5,–2,5) and ẩ3 = (−1,2,3). Express your answer in terms of the named vectors. Your answer
should be in the form 4ả₁ + 5ả₂ + 6ẩ3, which would be entered as 4a1 + 5a2 + 6a3.
(-4,-8, 6) =
-3a1+a2+2a3
b. Represent the vector (-4,-8,6) in terms of the ordered basis = {(1, −3,−2), (-5, -2,5),(-1,2,3)}. Your answer should be a vector of the general form .
[(-4,-8,6)] =
2.
Write 3
as a linear combination of the vectors
+
Express each column vector of BB as a linear combination of
the column vectors c1, c2, and c3 of B.
4 -2 4
B = |0
1
4
5
6
5
Enter first column as a linear combination of columns of B in
terms of the vectors c1, c2, and c3:
Enter second column as a linear combination of columns of B in
terms of the vectors c1, c2, and c3:
Enter third column as a linear combination of columns of B in
terms of the vectors c1, c2, and c3:
Chapter 2 Solutions
Linear Algebra With Applications (classic Version)
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