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Let $A=[a_{in}]$ be $m\times n$. For some $x=[x_j]\in\mathbb{F}^n$, then
$$\begin{align*}
Ax&=\begin{bmatrix}a_{11}x_1+\dots + a_{1n}x_n\end{bmatrix}
\end{align*} \\
&=x_1c_1+\dots + x_nc_n$$ where $c_j$ is the $j^{\text{th}}$ column of $A$. We say that $\text{span }\{c_1, \dots, c_n\}$ is the column space of $A$.
Let $r_1, \dots, r_m$ denote the rows of $A$. Then $\text{span }\{r_1, \dots, r_n\}$ is the row space of $A$.
Let $r_1, \dots, r_m$ denote the rows of $A$. Then $\text{span }\{r_1, \dots, r_n\}$ is the row space of $A$.
Concepts
$A$ is row equivalent to a matrix $R$, if and only if they have the same row space.Used In
Coming soonHypothesis
Coming soonResults
Coming soon
Let $A=[a_{in}]$ be $m\times n$. For some $x=[x_j]\in\mathbb{F}^n$, then
$$\begin{align*}
Ax&=\begin{bmatrix}a_{11}x_1+\dots + a_{1n}x_n\end{bmatrix}
\end{align*} \\
&=x_1c_1+\dots + x_nc_n$$ where $c_j$ is the $j^{\text{th}}$ column of $A$. We say that $\text{span }\{c_1, \dots, c_n\}$ is the column space of $A$.
Let $r_1, \dots, r_m$ denote the rows of $A$. Then $\text{span }\{r_1, \dots, r_n\}$ is the row space of $A$.
Let $r_1, \dots, r_m$ denote the rows of $A$. Then $\text{span }\{r_1, \dots, r_n\}$ is the row space of $A$.
Concepts
$A$ is row equivalent to a matrix $R$, if and only if they have the same row space.Used In
Coming soonHypothesis
Coming soonResults
Coming soon
FullPage
definition
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used in
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