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Subspaces
Defn. A subspace of a vector space V is a subset of V that is a vector space in its own right, using the same operations.
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Subspaces and the Three Matrix Spaces Subspaces Defn. A subspace of - - PowerPoint PPT Presentation
Subspaces and the Three Matrix Spaces Subspaces Defn. A subspace of - - PowerPoint PPT Presentation
Subspaces and the Three Matrix Spaces Subspaces Defn. A subspace of a vector space V is a subset of V that is a vector space in its own right, using the same operations. spaceTWO: 2 Checking Whether a Subspace ALGOR C ONDITIONS FOR S TO BE S
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Checking Whether a Subspace
ALGOR CONDITIONS FOR S TO BE SUBSPACE. (0) S contains the zero vector; (1) S is closed under addition; (2) S is closed under scalar multiplication.
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Examples: Polynomials and Functions
The degree-bounded polynomial space Pn is a subspace of the space P of all polynomials. And P is a subspace of the space C[t] of contin- uous functions. The set of continuous functions such that ∞
−∞ f(t)dt = 0 is a subspace of C[t].
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The Two Trivial Subspaces
Fact. The set containing just the zero-vector is always a subspace. The whole space is always a subspace of itself. Why?
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Subspaces and Geometry
- Fact. Every subspace of R3 is either {0}, a line
through the origin, a plane through the origin, or the space itself.
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Not a Subspace
≻ Consider the set of points (x, y) in R2 such that |x| = |y|. This is not a subspace: not closed under addition. ≻ Consider the set of points (x, y) in R2 such that x, y ≥ 0. This is not a subspace: not closed under scalar multiplication.
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Spans are Subspaces
Fact. If S is a set of vectors, then Span S is a subspace.
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The Null Space of a Matrix
Defn. The null space of matrix A, denoted Nul A, is all solutions to the homogeneous sys- tem Ax = 0. That is, all vectors mapped to 0 by the matrix transform x → Ax. If A is an m × n matrix, then Nul A is a vector space, and is a subspace of Rn.
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The Column Space of a Matrix
- Defn. The column space of matrix A, denoted
Col A, is all linear combinations of columns of A. If A is an m × n matrix, then Col A is a vector space, and is a subspace of Rm.
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The Row Space of a Matrix
Defn. The row space of matrix A, denoted Row A, is the set of linear combinations of rows
- f A.
If A is an m × n matrix, then Row A is a vector space, and is a subspace of Rn. Fact. If two matrices are row equivalent, then they have the same row space.
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Summary
A subspace of a vector space is a subset that is a vector space in its own right, using the same
- perations. To check whether a subset is a sub-
space, verify that it contains the zero vector, it’s closed under addition, and it’s closed under scalar multiplication. The span of a set of vectors is a subspace. The set of just the zero-vector is a subspace. Each nontrivial subspace of R3 is a line through the
- rigin or a plane through the origin.
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Summary (cont)
The null space of matrix A is all solutions to the homogeneous system Ax = 0. The column space of matrix A is all linear combinations of columns of A. The row space of matrix A is the set of linear combinations of rows of A.
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