### Linear maps: Linear maps

### Recording linear maps

Let #L:V\to W# be a linear transformation between vector spaces. If the images # L (\vec{a}_1), \ldots , L( \vec{a}_n)# under #L# of the vectors #\vec{a}_1,\ldots,\vec{a}_n# are known, then the image of each linear combination of these vectors is also known because of the linearity:

\[

L (x_1 \vec{a}_1 + \cdots + x_n\vec{a}_n) =

x_1 L (\vec{a}_1) + \cdots + x_n L (\vec{a}_n)

\] In particular, the image of each vector under #L# can be determined once the images of a set of basis vectors of #V# are known. The following theorem is based on this observation.

Linear map determined by the image of a basis

Let #V# and # W# be finite-dimensional vector spaces, #\basis{\vec{v}_1,\ldots ,\vec{v}_n}# a basis for #V#, and #\vec{w}_1,\ldots,\vec{w}_n# a sequence of #n# vectors in #W#. Then there is exactly one linear mapping # L :V\rightarrow W# with the property that # L( \vec{v}_i)=\vec{w}_i# for #i=1,\ldots,n#.

In particular, each linear map #V\to W# is *determined* by a matrix.

The images of a basis also determine the image space of the linear map:

Image as spanned subspace

Consider a linear map # L :V\rightarrow W#. If #V=\linspan{\vec{a}_1,\ldots ,\vec{a}_n}#, then \[ \im{L}=\linspan{ L( \vec{a}_1),\ldots, L( \vec{a}_n)}\]

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