# Column space

### Column space

#### Lessons

The column space of a matrix $A$ is a subspace of $\Bbb{R}^n$.

Suppose the matrix $A$ is:

$A=[v_1\;v_2\; \cdots \;v_n ]$

where $v_1,v_2,\cdots,v_n$ are the columns of $A$. Then the column space of $A$ is the set of vectors in $C(A)$ which forms a linear combination of the columns of $A$.

To see if a vector $\vec{b}$ is in the column space of $A$, we need to see if $\vec{b}$ is a linear combination of the columns of $A$. In other words,
$\vec{b} =x_1 v_1+x_2 v_2+\cdots+x_n v_n$

where $x_1,x_2,\cdots,x_n$ are solutions to the linear equation.

To find a basis for the column space of a matrix A, we:
1) Row reduce the matrix to echelon form.
2) Circle the columns with pivots in the row-reduced matrix.
3) Go back to the original matrix and circle the columns with the same positions.
4) Use those columns to write out the basis for $C(A)$.

Note that the vectors in the basis are linearly independent.
• Introduction
Column Space Overview:
a)
definition of the column space
$C(A)=$ column space
• A set of vectors which span{$v_1,v_2,\cdots,v_n$}
$\vec{b} \;\epsilon \;C(A),$ $\vec{b}= x_1 v_1+x_2 v_2+\cdots+x_n v_n$

b)
A vector in the column space
$\vec{b} =x_1 v_1+x_2 v_2+\cdots+x_n v_n$
• Changing to $Ax=b$ and solve

c)
Finding a basis for the column space
• Row reduce the matrix to echelon form.
• Locate the columns with pivots in the row-reduced matrix.
• Go back to the original matrix and find the columns with the same position.
• Use those columns to write out the basis

• 1.
Finding if a vector is in the column space
Let and . Determine whether $b$ is in the column space of $A$.

• 2.
Let and . Determine whether $b$ is in the column space of $A$.

• 3.
Finding a Basis for the Column Space
Here is the matrix $A$, and an echelon form of $A$. Find a basis for $C(A)$ (column space of $A$).

• 4.
Find a basis for the column space of $A$ if:

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5.
Subspace of $\Bbb{R}^n$
5.1
Properties of subspace
5.2
Column space
5.3
Null space
5.4
Dimension and rank