Unformatted text preview: M3710: Linear Algebra
Instructor: Phoebe McLaughlin Fall 2011
Homework 8 Solution (4.1) 4.1 Vector Spaces and Subspaces Selected assignment: 6, 8, 10, 14, 16, 18, 26, 32
6. The zero polynomial
cannot be written in the format of
That is, the zero polynomial is not in the set
subspace of .
8. Let
. Prove that is a subspace of
Proof.
1) Consider the zero polynomial
, since
2) For any
and
.
Since
,
For any
,
,
.
Hence is a subspace of . 14. for any
. Hence .
.
. .
is not a M3710: Linear Algebra
Instructor: Phoebe McLaughlin Fall 2011
Homework 8 Solution (4.1) 32a. Let and be subspaces of a vector space . Prove that
is a subspace of .
Proof.
1) Both and contain the zero vector of because they are subspaces of . Thus
.
2) For any
and
, respectively.
Since is a subspace of ,
.
Likewise
and is a subspace of
.
Thus
.
For any
,
and
, since
are subspaces of .
Thus
.
Therefore,
is a subspace of .
32b. Give an example in
subspace. to show that the union of two subspaces is not, in general, a Example:
Let
Then
But
Thus and . Then both and and are subspaces of .
. That is, is not a subspace of . is not closed under vector addition. . M3710: Linear Algebra
Instructor: Phoebe McLaughlin Fall 2011
Homework 8 Solution (4.1) ...
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 Spring '10
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