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.

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:

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 $\left(2\cdot x+5\right)^2$ in the vector space of all polynomials in $x$ of degree at most $2$ relative to the basis $\basis{1,x,x^2}$ .

$\rv{25,20,4}$

Working out the brackets yields

${\left(2\cdot x+5\right)^2} = 4\cdot x^2+20\cdot x+25$ 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 $\rv{25,20,4}$ .

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