True or False? In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. ( a ) If W is a subspace of a vector space V , then it has closure under scalar multiplication as defined in V . ( b ) If V and W are both subspaces of vector space U , then the intersection of V and W is also a subspace. ( c ) If U , V , and W are vector spaces such that W is a subspace of V and U is a subspace of V , then W = U .
True or False? In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. ( a ) If W is a subspace of a vector space V , then it has closure under scalar multiplication as defined in V . ( b ) If V and W are both subspaces of vector space U , then the intersection of V and W is also a subspace. ( c ) If U , V , and W are vector spaces such that W is a subspace of V and U is a subspace of V , then W = U .
Solution Summary: The author explains that if W is a subspace of vector space V, it has closure under scalar multiplication.
True or False?In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.
(
a
)
If
W
is a subspace of a vector space
V
, then it has closure under scalar multiplication as defined in
V
.
(
b
)
If
V
and
W
are both subspaces of vector space
U
, then the intersection of
V
and
W
is also a subspace.
(
c
)
If
U
,
V
, and
W
are vector spaces such that
W
is a subspace of
V
and
U
is a subspace of
V
, then
W
=
U
.
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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