Linear Combinations

A linear combination of a set of vectors uses the two operations of vector addition and vector scaling. Given a set of vectors, a linear combination of the set involves first scaling the vectors and then taking their sum.

Definition:
If \( \overrightarrow{w} \) is a vector in a vector space \( V \), then \( \overrightarrow{w} \) is said to be a linear combination of the vectors \( \overrightarrow{v_1}, \overrightarrow{v_2}, \dotsc , \overrightarrow{v_r} \) in \( V \) if \( \overrightarrow{w} \) can be expressed in the form

\( w = k_1\overrightarrow{v_1} + k_2\overrightarrow{v_2} + \dotsc + k_r\overrightarrow{v_r}\)

Where \( k_1 , k_2 ,\dotsc, k_r \) are scalars. These scalars ae called the coefficients of the linear combination.

Given two vectors \( \color{vred}\overrightarrow{v} \) & \( \color{vblue}\overrightarrow{w} \) a linear combination of them takes the following general form.

\( {\color{vred}a\overrightarrow{v}} + {\color{vblue}b\overrightarrow{w}} = \color{vgreen}\overrightarrow{z} \)

\( {\color{vred} a\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}} + \color{vblue} b\begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} = \color{vgreen} \begin{bmatrix} z_1 \\ z_2 \\ \vdots \\ z_n \end{bmatrix} \)

\( \color{vgreen}\overrightarrow{z} = \begin{bmatrix} {\color{vred}av_1} + {b\color{vblue}w_1} \\ {\color{vred}av_2} + {\color{vblue}bw_2} \\ \vdots \\ {\color{vred}av_n} + {\color{vblue}bw_n} \end{bmatrix} \)

Example:

Consider the four following vectors

\( \overrightarrow{v} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} \quad \overrightarrow{w} = \begin{bmatrix} 4 \\ 2 \end{bmatrix} \quad \overrightarrow{x} = \begin{bmatrix} 6 \\ 3 \end{bmatrix} \quad \overrightarrow{y} = \begin{bmatrix} 3 \\ 1 \end{bmatrix} \)

Vector \( \overrightarrow{x} \) is said to be a linear combination of \( \overrightarrow{v} \) and \( \overrightarrow{w} \) because it can be expressed in the following manner:
\( \overrightarrow{x} = 2\overrightarrow{v} + \frac{1}{2}\overrightarrow{w}\)

Vector \( \overrightarrow{y} \) is not a linear combination of \( \overrightarrow{v} \) and \( \overrightarrow{w} \) because there are no scalars \( k_1 \) and \( k_2 \) such that \( \overrightarrow{y} = k_1\overrightarrow{v} + k_2\overrightarrow{w}\)

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

The visualization shown to the right shows the linear combination of the two vectors v and w by randomly generated scalars. Every iteration a random pair of scalars a and b are generated. The animation first scales v and w by their respective scalars and then takes their sum.

Each line is a linear combination of v and w.

\( {\color{vred}a\overrightarrow{v}} + {\color{vblue}b\overrightarrow{w}} = \color{vgreen}\overrightarrow{r} \)

\( \color{vred}\overrightarrow{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \)

\( \color{vblue}\overrightarrow{w} = \begin{bmatrix} -1 \\ 0 \end{bmatrix} \)

\( \color{vred}a = \)

\( \color{vblue}b = \)

Try it Yourself

You can use the interactive tool below to visualize adding two vectors.

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Vector Space

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Linear Combinations

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