Linear Algebra 2.8 - 2.9 Subspace, dimension and rank

Section 2.8 : Subspace of R^n

A subspace of $\R^n$ if any set $H$ in $\R^n$ that has 3 properties:

The zero vector is in $H$
For each $u$ and $v$ in $H$ , then $u + v$ is in $H$
For each $u$ in $H$ and a scalar $c$ , then $cu$ is in $H$

Column Space

The Column space of a matrix $A$ is the set $\text{Col}\; A$ of all linear combinations of the columns of $A$ .

If $A=[a_1 \cdots a_n]$ , with the columns in $\R^m$ , then $\text{Col}\; A$ is the same as $\text{Span}\{a_1,\dots,a_n\}$ .

The column space of an $m\times n$ matrix is a subspace of $\R^m$ .

Null Space

The null space of a matrix $A$ is the set $\text{Null}\;A$ of all solutions of the homogenous equation $Ax=0$

The null space of an $m\times n$ matrix $A$ is a subspace of $\R^n$

Basic for a Subspace

A basic of a subspace $H$ of $\R^n$ is a linearly independent set in $H$ that spans $H$ .

The set $\{e_1,e_2,\dots,e_n\}$ is called the standard basic for $\R^n$ .

The pivot columns of matrix $A$ form a basis for the column space of $A$ .

Warning:

Be careful to use pivot columns of $A$ itself for the basis of $\text{Col}\; A$ . DO NOT use the columns of an echelon form!!!

Section 1.9 Dimension and Rank

Coordinate System

Suppose the set $B={b_1,\dots,b_p}$ is a basis for a subspace $H$ . For each $x$ in $H$ , the coordinates of $x$ relative to the basis $B$ are the weights $c_1,\dots,c_p$ such that $x=c_1b_1+\cdots+c_pb_p$ , and the vector in $\R^p$ :

is called the coordinate vector of $x$ (relative to $B$ ) or the B-coordinate vector of $x$ .

The Dimension of A Subspace

The dimension of a nonzero subspace $H$ , denoted by dim H, is the number of vectors in any basis for $H$ . The dimension of the zero subspace $\{0\}$ is defined to be zero.

The rank of a matrix $A$ , denoted by rank A, is the dimension of the column space of $A$ . 2

The Rank Theorem

If a matrix $A$ has $n$ columns, then $\text{rank A+dim Nul A}=n$ .

The Basic Theorem

Let $H$ be a p-dimensional subspace of $\R^n$ .

Any linearly independent set of exactly $p$ elements in $H$ is automatically a basis for $H$ .
Any set of $p$ elements of $H$ that spans $H$ is automatically a basis for $H$ .