### Linear maps: Matrices of Linear Maps

### Coordinates

*Earlier* we saw that linear images of #\mathbb{R}^n# to #\mathbb{R}^m# can be described using matrices. Thanks to the theorem *Linear map determined by the image of a basis*, such a description is also possible for other vector spaces, but then the description requires the choice of a basis and the use of coordinates.

Coordinates

In an #n#-dimensional vector space #V# we choose a basis #\alpha=\basis{\vec{a}_1,\ldots,\vec{a}_n}#. Then each vector #\vec{x}\in V# can be written uniquely as

\[

\vec{x}=x_1\vec{a}_1+x_2\vec{a}_2+\cdots +x_n\vec{a}_n

\] The numbers #x_1,\ldots ,x_n# are called the **coordinates** of #\vec{x}# relative to the basis #\alpha# and #\rv{x_1,\ldots ,x_n}# is called the **coordinate vector** (or more precisely the #\alpha#-**coordinate vector)** of #\vec{x}# relative to the basis #\alpha#.

*Previously* we saw that relative to #\alpha# the coordinates of #\vec{x}+ \vec{y}# are the sum of the coordinates of #\vec{x}# and #\vec{y}#, and that the coordinates of #\lambda\vec{x}# are exactly #\lambda# times the coordinates of #\vec{x}#. That means:

Coordinatization If we choose a basis #\alpha# in an #n#-dimensional vector space #V#, then the map that associates with each vector #\vec{x}\in V# its coordinates relative to the basis #\alpha# is an invertible linear map from #V# to #\mathbb{R}^n#.

We will usually denote this map by #\alpha# as well.

If #\alpha: V\to\mathbb{R}^n# is the coordinatization, then the corresponding basis of #V# is

\[\basis{\alpha^{-1}(\vec{e}_1),\ldots,\alpha^{-1}(\vec{e}_n)}\]

where \(\basis{\vec{e}_1,\ldots,\vec{e}_n}\) is the standard basis of #\mathbb{R}^n#.

Working out the brackets yields

\[ {\left(2\cdot x+3\right)^2} = 4\cdot x^2+12\cdot x+9\] The coordinate vector of this polynomial with respect to the basis #\basis{1,x,x^2}# consists of the coefficients of #1#, #x#, and #x^2#, and is thus #\rv{9,12,4}#.

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