Definition.
Let S be a subspace of Rn. A set B of vectors is a basis for S if
1. Span(B) = S,
2. B is linearly independent.
Fact 1. A basis of S is a largest collection of linearly independent vectors in S.
Fact 2. A basis of S is a smallest collection of vectors spanning S. Fact 3. All bases of S have the same number of vectors.
Row-reduction method to find a basis.
Let S = Span({u1,u2,...,uk}).
-Arrange uis as rows of a matrix, call it A.
-Find rref(A).
-The set of nonzero rows of rref(A) forms a basis of S.
Let S be a subspace of Rn. A set B of vectors is a basis for S if
1. Span(B) = S,
2. B is linearly independent.
Fact 1. A basis of S is a largest collection of linearly independent vectors in S.
Fact 2. A basis of S is a smallest collection of vectors spanning S. Fact 3. All bases of S have the same number of vectors.
Row-reduction method to find a basis.
Let S = Span({u1,u2,...,uk}).
-Arrange uis as rows of a matrix, call it A.
-Find rref(A).
-The set of nonzero rows of rref(A) forms a basis of S.
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| Example |
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