Linear Combinations

What is it?

Linear combinations are used in Linear Algebra to determine if a vector, or a matrix, is the result of the sum of products by a constant. For example, a linear combination of $x$ and $y$ would be any expression of the form $ax+by$, where $a,b$ are constants.

For example, given vectors $\mathbf{u}=(2,3)$ and $\mathbf{v}=(-1,4)$, we could get another vector $\mathbf{w}$ with the following equation $\mathbf{w}=2u+5v$. This results in a linear combination, and we can say that $\mathbf{w}$ is a linear combination of the vectors $\mathbf{u},\mathbf{v}$.

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Discovering if a vector is a linear combination

Given $\mathbf{w}=(13,22)$, we can verify if it is a linear combination from the other vectors $\mathbf{u}=(3,-2)$ and $\mathbf{v}=(1,7)$. We can use the general form of a linear combination:

$$\begin{array}{rl}& \mathbf{w}=ax+by\\ & \mathbf{w}=a\mathbf{u}+b\mathbf{v}\\ & (13,22)=a(3,-2)+b(1,7)\\ & \\ & \text{which can now be rewritten as}\\ & \\ & \{\begin{array}{l}3a+b=13\\ -2a+7b=22\end{array}\\ & \end{array}$$

Now we can take the linear system and solve it normally, as a matrix problem, with the Gauss-Jordan Elimination Method or the Gaussian Elimination Method.