Let V be a vector space with operations
and
and and let W be a non-empty
set of vectors of V. If W is a vector space with respect to the operations
and
in V, then W is called a subspace of V.
Sec 4.3
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Let V be a vector space with operations
and
and and let W be a non-empty
set of vectors of V. If W is a vector space with respect to the operations
and
in V, then W is called a subspace of V.
A subspace can be thought of as a vector space W that is a part of a
larger vector space V
4.3.1
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11:23 PM
Section4dot3 Page 2

Let V be a vector space with operations
and
and and let W be a non-empty
set of vectors of V. If W is a vector space with respect to the operations
and
in V, then W is called a subspace of V.
A subspace can be thought of as a vector space W that is a part of a
larger vector space V
The elements in W are already vectors since they come from the known
vector space V.
And W inherits the vector space operations
and
from V.
As a result, to check whether a set W is a vector space, it suffices to check
only the closure properties.
4.3.2
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Let V be a vector space with operations
and
and and let W be a non-empty
set of vectors of V. If W is a vector space with respect to the operations
and
in V, then W is called a subspace of V.
A subspace can be thought of as a vector space W that is a part of a
larger vector space V
The elements in W are already vectors since they come from the known
vector space V.
And W inherits the vector space operations
and
from V.
As a result, to check whether a set W is a vector space, it suffices to check
only the closure properties.
If W is closed under
and
, W inherits the algebraic and inverse
properties of
V, so all other 8 properties of a vector space are satisfied.
4.3.3
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11:23 PM
Section4dot3 Page 4