Basis Set

Just to give you a vague rough idea, you know how you can write any word in the English language using the 26 letters of the alphabet?

Assume we are interested in a real 3D vector space. We can express any vector as a linear combination of three vectors if we choose the three vectors correctly. Our vector will be (a,b,c) where a,b and c are real values.

It is typical to choose {1,0,0}, {0,1,0} and {0,0,1}. With these being our choices the linear combination for {a,b,c} is as follows:

\begin{bmatrix}a \\b \\c \end{bmatrix} = a \begin{bmatrix}1 \\0 \\0 \end{bmatrix} + b \begin{bmatrix}0 \\1 \\0 \end{bmatrix} + c \begin{bmatrix}0 \\0 \\1 \end{bmatrix}

We would like to show you something. What if we replaced the “1” values with values of 2.54?

\begin{bmatrix}a \\b \\c \end{bmatrix} = a' \begin{bmatrix}2.54 \\0 \\0 \end{bmatrix} + b' \begin{bmatrix}0 \\2.54 \\0 \end{bmatrix} + c' \begin{bmatrix}0 \\0 \\2.54 \end{bmatrix}

latex]\begin{bmatrix}2.54 \\ 5.08 \\ 5.08 \end{bmatrix} = 1 \begin{bmatrix}2.54 \\0 \\0 \end{bmatrix} + 2 \begin{bmatrix}0 \\2.54 \\0 \end{bmatrix} + 2 \begin{bmatrix}0 \\0 \\2.54 \end{bmatrix}[/latex

  • (a,b,c)=(2.54, 5.08, 5.08)
  • (a’,b’,c’)=(1,2,2)
  • The first vector is in centimeters and the second vector is in inches