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 UBSPACE . (0) S contains the zero vector; (1) S is closed under addition; (2) S is closed under scalar multiplication. spaceTWO: 3

Examples: Polynomials and Functions The degree-bounded polynomial space P n 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 ] . spaceTWO: 4

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? spaceTWO: 5

Subspaces and Geometry Fact. Every subspace of R 3 is either { 0 } , a line through the origin, a plane through the origin, or the space itself. spaceTWO: 6

Not a Subspace ≻ Consider the set of points ( x, y ) in R 2 such that | x | = | y | . This is not a subspace: not closed under addition. ≻ Consider the set of points ( x, y ) in R 2 such that x, y ≥ 0 . This is not a subspace: not closed under scalar multiplication. spaceTWO: 7

Spans are Subspaces Fact. If S is a set of vectors, then Span S is a subspace. spaceTWO: 8

The Null Space of a Matrix Defn. The null space of matrix A , denoted Nul A , is all solutions to the homogeneous sys- tem A x = 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 R n . spaceTWO: 9

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 R m . spaceTWO: 10

The Row Space of a Matrix Defn. The row space of matrix A , denoted Row A , is the set of linear combinations of rows of A . If A is an m × n matrix, then Row A is a vector space, and is a subspace of R n . Fact. If two matrices are row equivalent, then they have the same row space. spaceTWO: 11

Summary A subspace of a vector space is a subset that is a vector space in its own right, using the same operations. 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 R 3 is a line through the origin or a plane through the origin. spaceTWO: 12

Summary (cont) The null space of matrix A is all solutions to the homogeneous system A x = 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 . spaceTWO: 13