basis for a subspace definition - university of houston · section 2.9 dimension of a subspace...
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Basis for a Subspace
Definition:
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Question: How can we use the RREF to find a basis for the column space of a matrix?
Example: Find a basis for the column space of
1 1 1 3
2 1 3 2
1 2 4 1
3 0 2 5
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Example: Find bases for the null space and column space of
1 1 1 3
2 1 3 2
1 2 4 1
3 0 2 5
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Example: Show that 111
,121
,212
is a basis for
. (Hint: Recall the special situation. n vectors give a basis for if and only if they are linearly independent.)
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Example: Find a basis for the null space of the matrix
1 3 0 1 1
1 2 1 2 3
Also give a basis for the column space of the matrix.
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Example: Find a basis for the subspace spanned by the vectors
1123
,
2314
,
0132
,
1477
,
3769
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Section 2.9
Dimension of a Subspace
Theorem: (The Basis Theorem) Any two basis for a subspace H of have the same number of elements. Furthermore, if a basis for H has p elements, then any linearly independent subset of H with p elements is a basis for H, and any subset of H with p entries that spans H is a basis for H.
Corollary: A subset of is a basis for if and only if it consists of linearly independent vectors.
Terminology: Standard basis for .
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Example: Show that
2 32
2 2| , , ∈
is a subspace of . Give a basis for , and the dimension of .
Finally, determine whether 111
∈ .
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Nullity and Rank
Theorem: (rank theorem) If is an matrix, then
rank A + nullity A = n
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Rank and the Invertible Matrix Theorem
Theorem: (The Invertible Matrix Theorem) Suppose A is an matrix. Then the following statements are equivalent to
A being an invertible matrix.
m. The columns of form a basis for . n. Col o. dimCol p. rank q. Nul 0 r. dimNul 0
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Example: A basis for the null space of a 2 by 3 matrix A is given by
1
1
2
Give the rank of A.
Example: The RREF of the matrix A is
1 1 0
0 0 1
0 0 0
Give the rank of A, the nullity of A, and a basis for the null space of A.
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B-Coordinates of a Vector x in a Subspace H with Respect to a Basis B for H
Definition:
Example: Show that 21, 11
is a basis for . Then
give 11
.
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Exam
mple: Prooblem 30
0 in the ttextbookk.
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