Coordinates

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#.

The map that adds to each vector #\vec{x}# of #V# the associated coordinate vector in #\mathbb{R}^n# is a bijection. Of course the coordinates depend on the chosen 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#.

The map #\alpha# is invertible, and sends the #i#-th basis vector of #\alpha# to #\vec{e}_i#. The inverse map thus sends #\vec{e}_i# to the #i#-th basis vector of #\alpha#.

Denoting both the basis and the map by #\alpha# rarely causes confusion: #\alpha(\vec{x})# is the coordinate vector of the vector #\vec{x}\in V# with respect to the basis #\alpha#.

Give the coordinate vector of #\left(3\cdot x+7\right)^2# in the vector space of all polynomials in #x# of degree at most #2# relative to the basis #\basis{1,x,x^2}#.

#\rv{49,42,9}#

Working out the brackets yields
\[ {\left(3\cdot x+7\right)^2} = 9\cdot x^2+42\cdot x+49\] 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{49,42,9}#.

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