3.5 Subspaces of R^p
Readings
Try the Preview Activity and read Chapter 3 Section 5 in Understanding Linear Algebra by David Austin.
A subspace of [latex]\mathbb{R}^p[/latex] is a subset of vectors [latex]H[/latex] such that any linear combination of vectors in [latex]H[/latex] is also in [latex]H[/latex]
Applying the subspace definition:
A basis of a subspace [latex]H \subset \mathbb{R}^p[/latex] is a set of vectors in [latex]H[/latex] that are linearly independent and span [latex]H[/latex]. Any two bases of a subspace contain the same number of vectors. This number is called the dimension of the subspace, and is written [latex]dim(H)[/latex]
Video
Overview:
Subspaces of [latex]\mathbb{R}^p[/latex]
If A is an mxn matrix then, we can think of a matrix transformation [latex]T: \mathbb{R}^n \to \mathbb{R}^m[/latex], with [latex]T(\vec{x}) = A\vec{x}[/latex].
The Null space of A, Nul(A) is a subspace of the domain, [latex]\mathbb{R}^n. Nul(A) = \{\vec{x}| A\vec{x} =\vec{0}.\}[/latex]
Finding Nullspaces Example:
The Column space of A, Col(A) is a subspace of the codomain, [latex]\mathbb{R}^m[/latex]. It is the range of the matrix transformation T. [latex]Col(A) = \{\vec{y}| A\vec{x} =\vec{y}, \mbox{for some} \vec{x}\in \mathbb{R}^n \}[/latex]
Finding Column Spaces Example: