Let W be a nonzero subspace of R n . Show that W has a basis. [ Hint: Let w 1 be any nonzero vector in W If { w 1 } is a spanning set for W , then we are done. If not, there is a vector w 2 in W such that { w 1 , w 2 } is linearly independent. Why? Continue by asking whether this is a spanning set for W . Why must this process eventually stop?]
Let W be a nonzero subspace of R n . Show that W has a basis. [ Hint: Let w 1 be any nonzero vector in W If { w 1 } is a spanning set for W , then we are done. If not, there is a vector w 2 in W such that { w 1 , w 2 } is linearly independent. Why? Continue by asking whether this is a spanning set for W . Why must this process eventually stop?]
Solution Summary: The author explains that if W is a nonzero subspace of Rn, then it has the basis.
Let
W
be a nonzero subspace of
R
n
. Show that
W
has a basis. [Hint: Let
w
1
be any nonzero vector in
W
If
{
w
1
}
is a spanning set for
W
, then we are done. If not, there is a vector
w
2
in
W
such that
{
w
1
,
w
2
}
is linearly independent. Why? Continue by asking whether this is a spanning set for
W
. Why must this process eventually stop?]
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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