Vector spaces

Here is a program that depicts the familiar Cartesian coordinate system in three dimensions. The unit vectors along the principal axes are represented by red (x), green (y), and blue (z) arrows. The grid lines appear at one-unit intervals. If you enter three comma-separated values in the text field and then press Go, a sphere is placed at the current coordinates.

Plotting points

Viewed as a vector space, $R^3$ is spanned by three basis vectors $e_1$ , $e_2$ , and $e_3$ . These are depicted by the three arrows. Any vector $v\in R^3$ can be expressed uniquely as the linear combination of basis vectors: $v = x_1e_1+x_2e_2+x_3e_3$ . When the basis vectors are assumed, we can write vector $v$ more compactly as an ordered tuple $(x_1,x_2,x_3)$ of real coefficients. For example $(2,3,-4)=2e_1+3e_2-4e_3$ . Regarding the basis vectors, $e_1=(1,0,0)$ since $e_1 = 1\times e_1+ 0\times e_2+ 0\times e_3$ ; similarly, $e_2=(0,1,0)$ and $e_3=(0,0,1)$ .

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Michael Laszlo

It is not a something, but not a nothing either

Vector spaces