Linear Algebra | CHAPTER3 / Computing Linear Maps

The prior section shows that a linear map is determined by its action on a basis. In fact, the equation

$\begin{equation*} h(\vec{v}) =h(c_1\cdot\vec{\beta}_1+\dots+c_n\cdot\vec{\beta}_n) =c_1\cdot h(\vec{\beta}_1)+\dots +c_n\cdot h(\vec{\beta}_n) \tag*{}\end{equation*}$

shows that, if we know the value of the map on the vectors in a basis, then we can compute the value of the map on any vector $\vec{v}$ at all. We just need to findi the $c$ ‘s to express $\vec{v}$ with respect to the basis.

This section gives the scheme that computes, from the representation of a vector in the domain

$\rep{\vec{v}}{B}$ , the representation of that vector’s image in the codomain $\rep{h(\vec{v})}{D}$ ,

using the representations of $h(\vec{\beta}_1), \ldots, h(\vec{\beta}_n)$ .

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