Linear Combination of vectors

Given two vectors $\vec{u}$ and $\vec{v}$ we name linear combination of $\vec{u}$ and $\vec{v}$ to any expression of the form: $λ \vec{u} + μ \vec{v}$ where $λ$ and $μ$ are real numbers.

A vector $\vec{w}$ is a linear combination of $\vec{u}$ and $\vec{v}$ if real (scalar) numbers (escalars) $λ$ and $μ$ exist such that we can express $\vec{w}$ as follows: $\vec{w} = λ \vec{u} + μ \vec{v}$ .

The vectors we have been working with until now are vectors on the plane, so they have two components. In this case we can express any vector $\vec{w}$ as a linear combination of two non parallel vectors $\vec{u}$ and $\vec{v}$ . This combination is unique.

Example

Is the vector $\vec{w} = (- 1, 3)$ a linear combination of the vectors of $\vec{u} = (1, 2)$ and $\vec{v} = (0, 3)$ ?

We want to find $λ$ and $μ$ so as $\vec{w} = λ \vec{u} + μ \vec{v}$ . We have: $(- 1, 3) = λ (1, 2) + μ (0, 3) = (λ, 2 λ) + (0, 3 μ) = (λ, 2 λ + 3 μ)$

Therefore: $\begin{array}{rcl} - 1 & = & λ \\ 3 & = & 2 λ + 3 μ \end{array}} \Rightarrow λ = - 1, μ = \frac{5}{3}$

We have just found values for $λ$ and $μ$ for which $\vec{w} = λ \vec{u} + μ \vec{v}$ is true. So the answer is "yes", we can express $\vec{w} = (- 1, 3)$ as a linear combination of $\vec{u} = (1, 2)$ and $\vec{v} = (0, 3)$ .