3.5	Subspaces of R^p

Try the Preview Activity and read Chapter 3 Section 5 in Understanding Linear Algebra by David Austin.

A subspace of

R^{p}

is a subset of vectors

H

such that any linear combination of vectors in

H

is also in

H

Applying the subspace definition:

A basis of a subspace

H \subset R^{p}

is a set of vectors in

H

that are linearly independent and span

H

. Any two bases of a subspace contain the same number of vectors. This number is called the dimension of the subspace, and is written

d i m (H)

Overview:

Subspaces of $R^{p}$

If A is an mxn matrix then, we can think of a matrix transformation $T : R^{n} \to R^{m}$ , with $T (\vec{x}) = A \vec{x}$ .

The Null space of A, Nul(A) is a subspace of the domain,

R^{n} . N u l (A) = {\vec{x} | A \vec{x} = \vec{0} .}

Finding Nullspaces Example:

The Column space of A, Col(A) is a subspace of the codomain,

R^{m}

. It is the range of the matrix transformation T.

C o l (A) = {\vec{y} | A \vec{x} = \vec{y}, for some \vec{x} \in R^{n}}

Finding Column Spaces Example:

License

Icon for the Creative Commons Attribution 4.0 International License