Linear Combinations

What’s a linear combination?

If $\bold{x_1}$ and $\bold{x_2}$ are any two objects and $\alpha_1$ and $\alpha_2$ are any two scalars, then $\alpha_1\bold{x_1}+\alpha_2\bold{x_2}$ is a linear combination of $\bold{x_1}$ and $\bold{x_2}$. We can have several linear combinations of $\bold{x_1}$ and $\bold{x_2}$ by varying $\alpha_1$ and $\alpha_2$.

Example

Find any three linear combinations of $\bold{x_1} = \begin{bmatrix} 3\\ 5\\ 9 \end{bmatrix}$ and $\bold{x_2} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$.

Note: Objects like column-vectors, such as $\begin{bmatrix} 3\\ 5\\ 9 \end{bmatrix}$ , are commonly used to represent points in space. We’ll discuss the vectors and vector spaces in detail in upcoming lessons, but for now, it’s okay to simply view these objects as arrays.

Solution:

$\bold{y_1}=2\begin{bmatrix} 3 \\ 5 \\ 9 \end{bmatrix}+3 \begin{bmatrix} 1\\ 2\\ 1 \end{bmatrix}=\begin{bmatrix} 9\\ 16\\ 21 \end{bmatrix} \hspace{5mm} \text{where} \quad \alpha _1 = 2, \alpha _2 = 3$ ...