Basis of Vector Spaces

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Change of Basis

The same object, here a circle, can be completely different when viewed in other vector spaces.

Basic Description

A point in space can be located by giving a set of coordinates. These coordinates can be thought of as showing how much to multiply a certain set of vectors, known as basis vectors, to reach the point. For example, the so-called standard basis vectors for two-dimensional Euclidean Space are  \begin{bmatrix} 1 \\ 0\\ \end{bmatrix}, \begin{bmatrix} 0 \\ 1\\ \end{bmatrix} so the point (2,3) relative to this basis has us multiply the first basis vector by 2, the second by 3, then add the two vectors to reach our point. We say that the point (2,3) has coordinate vector  \begin{bmatrix} 2\\ 3\\ \end{bmatrix} relative to the standard basis. As another example, relative to the basis vectors  \begin{bmatrix} 2 \\ 0\\ \end{bmatrix}, \begin{bmatrix} 0 \\ 3\\ \end{bmatrix} the same point has coordinate vector  \begin{bmatrix} 1 \\ 1\\ \end{bmatrix} .

It is often useful to use basis vectors that are not simply Euclidean vectors. For example, polar coordinates use the basis vectors  r,\theta where  r represents distance from the origin and  \theta represents rotation angle from the positive x-axis. The point (0,1) has coordinate vector  \begin{bmatrix} 1 \\ \pi/2\\ \end{bmatrix} relative to these polar basis vectors.

This page's main image shows the coordinates of the points contained in a circle of a radius one relative to three different bases. The coordinates relative to the standard basis forms a circle, relative to the polar basis vector forms a rectangle, and relative to the basis vectors  \begin{bmatrix} 0.5 \\ 1\\ \end{bmatrix} forms an ellipse.

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