Product of an Associated Matrix and a Coordinate Vector

Topics: Algebra - Linear Transformation - Vector Space - Associated Matrix of a Linear Transformation - Coordinate Vector

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(theorem)

Let $V,W$ be vector spaces on $\mathbb{K}$, and $T:V\to W$ a linear transformation.

Let:

• $v\in V$
• $\beta =\{v_1,\dots ,v_n\}$ a basis for $V$
• $\gamma =\{w_1,\dots ,w_n\}$ a basis for $W$

Then:

$$[T{]}_{\beta }^{\gamma }[v{]}_{\beta }=[T(v){]}_{\gamma }$$

That is, the product of the associated matrix of $T$ and the coordinate vector of $v$ is equal to the coordinate vector of $T(v)$.

Example

Let $T:{\mathbb{R}}^3\to {\mathbb{R}}^2$ and…

$$[T{]}_{\beta }^{\gamma }=\left(\begin{array}{ccc}1& -1& 0\\ 2& 0& 0\\ 3& 4& 1\end{array}\right)$$

…where $\beta =\gamma =\{(1,0,0),(1,1,0),(1,1,1)\}$.

Let’s find $T(2,0,-1)$.

First, we find the coordinate vector (under $\beta$) of this element:

$$[(2,0,-1){]}_{\beta }=\left(\begin{array}{c}2\\ 1\\ -1\end{array}\right)$$

Then, we multiply:

$$[T{]}_{\beta }^{\gamma }[(2,0,-1){]}_{\beta }=\left(\begin{array}{ccc}1& -1& 0\\ 2& 0& 0\\ 3& 4& 1\end{array}\right)\left(\begin{array}{c}2\\ 1\\ -1\end{array}\right)$$

$$=\left(\begin{array}{c}1\\ 4\\ 9\end{array}\right)=[T(2,0,-1){]}_{\gamma }$$

Finally, with that, we can write:

$$T(2,0,-1)=1(1,0,0)+4(1,1,0)+9(1,1,1)$$

$$=(14,13,9)$$