Let W be the subspace described in Exercise 2. For each vector x that follows, determine if x is in W . If x is in W , then express x as a linear combination of the basis vectors found in Exercise 2. a) x = [ − 3 3 1 1 ] b) x = [ 0 3 2 − 1 ] c) x = [ 7 8 3 2 ] d) x = [ 4 − 2 0 − 2 ] In Exercises 1-8, let W be the subspace of R 4 consisting of vectors of the form x = [ x 1 x 2 x 3 x 4 ] Find a basis for W when the components of x satisfy the given conditions. 2. x 1 + x 2 − x 3 + x 4 = 0 x 2 − 2 x 3 − x 4 = 0
Let W be the subspace described in Exercise 2. For each vector x that follows, determine if x is in W . If x is in W , then express x as a linear combination of the basis vectors found in Exercise 2. a) x = [ − 3 3 1 1 ] b) x = [ 0 3 2 − 1 ] c) x = [ 7 8 3 2 ] d) x = [ 4 − 2 0 − 2 ] In Exercises 1-8, let W be the subspace of R 4 consisting of vectors of the form x = [ x 1 x 2 x 3 x 4 ] Find a basis for W when the components of x satisfy the given conditions. 2. x 1 + x 2 − x 3 + x 4 = 0 x 2 − 2 x 3 − x 4 = 0
Solution Summary: The author explains that the vector x is in W and its linear combination is expressed as (1).
Let
W
be the subspace described in Exercise 2. For each vector
x
that follows, determine if
x
is in
W
. If
x
is in
W
, then express
x
as a linear combination of the basis vectors found in Exercise 2.
a)
x
=
[
−
3
3
1
1
]
b)
x
=
[
0
3
2
−
1
]
c)
x
=
[
7
8
3
2
]
d)
x
=
[
4
−
2
0
−
2
]
In Exercises 1-8, let
W
be the subspace of
R
4
consisting of vectors of the form
x
=
[
x
1
x
2
x
3
x
4
]
Find a basis for
W
when the components of
x
satisfy the given conditions.
2.
x
1
+
x
2
−
x
3
+
x
4
=
0
x
2
−
2
x
3
−
x
4
=
0
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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