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I'm having trouble understanding this. Why does the column space of the matrix of a linear transformation equal the image of the linear transformation?

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Look at a simple concrete example, say the matrix $$A=\begin{bmatrix}2&5&-1\\3&-1&2\end{bmatrix}\;.$$ The column space of $A$ is by definition the set of all linear combinations of the columns of $A$, i.e., the set of all vectors of the form

$$\alpha{2\brack 3}+\beta{5\brack -1}+\gamma{-1\brack 2}$$

for real numbers $\alpha,\beta$, and $\gamma$.

Now, what vectors are in the image of $A$? The image of $A$ consists of all vectors of the form $Av$, where $v$ is a $3\times 1$ column vector. A typical $3\times 1$ vector is $$v=\begin{bmatrix}\alpha\\\beta\\\gamma\end{bmatrix}\;,$$ and $$Av=\begin{bmatrix}2&5&-1\\3&-1&2\end{bmatrix}\begin{bmatrix}\alpha\\\beta\\\gamma\end{bmatrix}=\begin{bmatrix}2\alpha+5\beta-\gamma\\3\alpha-\beta+2\gamma\end{bmatrix}=\alpha{2\brack3}+\beta{5\brack-1}+\gamma{-1\brack2}\;.$$

Thus, both the column space of $A$ and the image of $A$ consist of all vectors of the form $$\alpha{2\brack3}+\beta{5\brack-1}+\gamma{-1\brack2}\;,$$ so they’re the same.

To see that this always happens, you have to recognize that a product $Av$ always simply forms a linear combination of the columns of $A$. If the columns of $A$ are $A_1,\dots,A_n$, and $$v=\begin{bmatrix}v_1\\\vdots\\v_n\end{bmatrix}\;,$$ then $$Av=v_1A_1+v_2A_2+\ldots+v_nA_n\;.$$ When you calculate the $k$-th entry in $Av$, for instance, you get $v_1a_{k1}+v_2a_{k2}+\ldots v_na_{kn}$, where $a_{ki}$ is the $k$-th entry in $A_i$, so $$Av=\begin{bmatrix}v_1a_{11}+v_2a_{12}+\ldots v_na_{1n}\\v_1a_{21}+v_2a_{22}+\ldots v_na_{2n}\\v_1a_{31}+v_2a_{32}+\ldots v_na_{3n}\\\vdots\\v_1a_{m1}+v_2a_{m2}+\ldots v_na_{mn}\end{bmatrix}\;,\tag{1}$$ assuming that $A$ is $m\times n$. But $(1)$ is just

$$Av=v_1\begin{bmatrix}a_{11}\\a_{21}\\a_{31}\\\vdots\\a_{m1}\end{bmatrix}+v_2\begin{bmatrix}a_{12}\\a_{22}\\a_{32}\\\vdots\\a_{m2}\end{bmatrix}+\ldots+\begin{bmatrix}a_{1n}\\a_{2n}\\a_{3n}\\\vdots\\a_{mn}\end{bmatrix}=v_1A_1+v_2A_2+\ldots+v_nA_n\;.$$ In short, every $Av$ in the image of $A$ is a linear combination of the columns of $A$ and vice versa.

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It's a simple observation that $$ Ax = A_1 x_1 + A_2 x_2 + \cdots +A_n x_n $$ where $A_i$ is the $i$-th column of $A$ and $x$ is a (column) vector.

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When you multiply a matrix $A$ by the $i$-th basis element, you get the $i$-th column of the matrix. Multiplying by an arbitrary vector (which is a linear combination of basis elements) gives you, by linearity, a linear combination of the columns of $A$. In other words, the image of $A$ is the set of linear combinations of its columns, which is its column space.

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