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− | === $C(A)$ === | + | === The Column Space $C(A)$ === |
Let $A$ be $m \times n$ matrix: | Let $A$ be $m \times n$ matrix: | ||
* $A = \left[ \mathop{a_1}\limits_|^| \, \mathop{a_2}\limits_|^| \ \cdots \ \mathop{a_n}\limits_|^| \right]$ | * $A = \left[ \mathop{a_1}\limits_|^| \, \mathop{a_2}\limits_|^| \ \cdots \ \mathop{a_n}\limits_|^| \right]$ | ||
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* $C(A)$ is in $\mathbb R^r$ space where $r \leqslant n$ is the $A$'s [[Rank (Matrix)|Rank]] | * $C(A)$ is in $\mathbb R^r$ space where $r \leqslant n$ is the $A$'s [[Rank (Matrix)|Rank]] | ||
* so the dimensionality is at most the number of columns, and at least the rank of the matrix | * so the dimensionality is at most the number of columns, and at least the rank of the matrix | ||
+ | |||
+ | |||
+ | <img width="50%" src="http://alexeygrigorev.com/projects/imsem-ws14-lina/img-svg/diagram1.svg" /> | ||
+ | |||
+ | |||
+ | It's also called the ''range'' of $A$: | ||
+ | * $\text{ran}(A) = \{ \mathbf y \in \mathbb R^m \ : \ \mathbf y = A \mathbf x \ \forall \mathbf x \in \mathbb R^n \}$ | ||
+ | * if we think about [[Linear Transformation]] $T_{A}$ formed by $A$, this is what it does to an $n$-dimensional vector $\mathbf x$: it brings it to an $m$-dinesional vector $\mathbf y$ | ||
+ | * all such vectors $\mathbf y$ form the column space of $A$ | ||
+ | |||
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For example, the following can solve it: | For example, the following can solve it: | ||
* $\mathbf 0_4$, because $\mathbf x = \mathbf 0_3$ will solve it | * $\mathbf 0_4$, because $\mathbf x = \mathbf 0_3$ will solve it | ||
− | * | + | * <math>\begin{bmatrix} |
1 \\ | 1 \\ | ||
2 \\ | 2 \\ | ||
3 \\ | 3 \\ | ||
4 \\ | 4 \\ | ||
− | \end{bmatrix} | + | \end{bmatrix}</math> - one of the columns, so <math>\mathbf x = \begin{bmatrix} |
1 \\ | 1 \\ | ||
0 \\ | 0 \\ | ||
0 \\ | 0 \\ | ||
− | \end{bmatrix} | + | \end{bmatrix}</math> |
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== Sources == | == Sources == | ||
* [[Linear Algebra MIT 18.06 (OCW)]] | * [[Linear Algebra MIT 18.06 (OCW)]] | ||
+ | * [[Matrix Computations (book)]] | ||
[[Category:Linear Algebra]] | [[Category:Linear Algebra]] |
A column space $C(A)$ of a matrix $A$ is a subspace formed by columns of $A$
Let $A$ be $m \times n$ matrix:
It's also called the range of $A$:
It's a subspace:
Suppose we have a matrix $A \in \mathbb R^{3 \times 2}$
$A = \begin{bmatrix} 1 & 3 \\ 2 & 3 \\ 4 & 1 \\ \end{bmatrix}$
Subspace from columns - $C(A)$ - the Column Space of $A$:
Column Space $C(A)$ of $A$ is important: the system $A \mathbf x = \mathbf b$ has the solution only when $\mathbf b \in C(A)$
For example:
$A = \begin{bmatrix} 1 & 1 & 2 \\ 2 & 1 & 3 \\ 3 & 1 & 4 \\ 4 & 1 & 5 \\ \end{bmatrix}$.
Since there are only 3 columns, in $A \mathbf x = \mathbf b$
For example, the following can solve it:
if $\mathbf b \not \in C(A)$ there's no way to solve the system
What's the dimension of $C(A)$?