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Linear Algebra test 2
material from 4.3
6
Mathematics
Undergraduate 2
10/25/2016
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Term
subspace of V
 Definition
a nonempty subset W of a vector space V, when W is a vector space under the operations of adition and scalar multiplication defined in V
Term
test for a subspace
 Definition

if W is a nonempty subset of a vector space V, then W is a subspace of V if and only if the following closure conditions hold:

1) if u and v are in W, the u+v is in W

2) if u is in W and c is a scalar, then cu is in W

note: must check that W is nonempty (i.e. W has at least one element in it). an easy way to do this is to check that it has the zero vector in W
Term
zero subspace
 Definition
subspace consisting of only the zero vector
Term
trivial subspace
 Definition

1) subspace containing itself

2) zero subspace
Term
proper subspace of V
 Definition

if subspace W is not a trivial subspace

i.e. w≠v and w≠{0(vector)}
Term
intersection of two subspaces is a subspace
 Definition
if v and w are both subspaces of a vector space u, then the intersection of v and w is also a subspace of u. the intersection is denoted by v∩w
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