What Is A Linear Combination?


Hi there. Here is a short post of the concept of a linear combination found in Linear Algebra.


Suppose we have vectors \boldsymbol{v_1}, \boldsymbol{v_2}, \dots, \boldsymbol{v_r} in \mathbb{R}^{n}} and scalars k_1, k_2, \dots k_r. The linear combination (and vector) \textbf{w} can be expressed as:

\displaystyle \boldsymbol{w} = k_{1} \boldsymbol{v_1} + k_{2} \boldsymbol{v_2} + k_{3} \boldsymbol{v_3} + \dots + k_{r} \boldsymbol{v_r}

where r refers to the number of vectors (terms) and n is the dimension of each vector. Each \boldsymbol{v_i} from i = 1, 2, … , r is of the form

\displaystyle \boldsymbol{v_i}= \begin{bmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{bmatrix}


Examples

Example One

A simple linear combination in \mathbb{R}^{2} is 6 \begin{bmatrix} x \ y \ \end{bmatrix} - 4 \begin{bmatrix} 2x \ 5y \ \end{bmatrix}. This would be equal to

\displaystyle \begin{bmatrix} 6x \\ 6y \\ \end{bmatrix} - \begin{bmatrix} 8x \\ 20y \\ \end{bmatrix} = \begin{bmatrix} 6x - 8x \\ 6y - 20y \\ \end{bmatrix} =\begin{bmatrix} -2x \\ -14y \\ \end{bmatrix}

Example Two

An example of a linear combination in \mathbb{R}^{3} would be 7 \begin{bmatrix} x \ y \ z\end{bmatrix} + 2 \begin{bmatrix} x \ -2y \ 3z\end{bmatrix} - \begin{bmatrix} x \ -y \ 8z\end{bmatrix}. This is equal to

\displaystyle \begin{bmatrix} 7x \\ 7y \\ 7z\end{bmatrix} + \begin{bmatrix} 2x \\ -4y \\ 6z\end{bmatrix} + \begin{bmatrix} -x \\ +y \\ -8z\end{bmatrix} = \begin{bmatrix} 7x + 2x - x \\ 7y - 4y + y \\ 7z + 6z -8z\end{bmatrix} = \begin{bmatrix} 8x \\ 4y\\ 5z \end{bmatrix}

The examples above are nice examples since we can apply vector operations. Sometimes we may not have scalars that are numbers; we would be stuck with k_1, k_2, \dots k_r.


Reference

Elementary Linear Algebra (Tenth Edition) by Howard Anton

The featured image is from http://www.euclideanspace.com/maths/geometry/space/vector/vec2dBases.png.

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