Linear Combination

A math object is a linear combination of a set if we can create the math object by summing together scaled elements of the set.

You will probably want to see an example…

The vector \begin{bmatrix} a \\ b \\ c \end{bmatrix} is a linear combination of

  • \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}

because we can get it through the following summation:

\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}

Whatever “type” the answer is, the elements in the set must be of the same type:

  • a vector would be a linear combination of vectors
  • a matrix would be a linear combination of matrices

This is because when you scale a math object you don’t change its type.

Appendix A

When we say “scaled math object” we refer to a math object that has been changed by multiplication by a scalar.

Some textbooks say multiplication by a constant.