Linear Combinations of Vectors – The Basics

In linear algebra, we define the concept of linear combinations in terms of vectors. But, it is actually possible to talk about linear combinations of anything as long as you understand the main idea of a linear combination:

(scalar)(something 1) + (scalar)(something 2) + (scalar)(something 3)

These “somethings” could be “everyday” variables like \(x\) and \(y\) (\(3x\) + \(2y\) is a linear combination of \(x\) and \(y\) for instance) or something more complicated like polynomials. In general, a linear combination is a particular way of combining things (variables, vectors, etc) using scalar multiplication and addition.


Working with vectors

Now back to vectors. Let’s say we have the following vectors:

\(\vec{v}_1 = \left[ \begin{array}{c}1\\ 2\\ 3\end{array} \right]\), \(\vec{v}_2 = \left[ \begin{array}{c}3\\ 5\\ 1\end{array} \right]\), \(\vec{v}_3 = \left[ \begin{array}{c}0\\ 0\\ 8\end{array} \right]\)

What would linear combinations of these vectors look like? Well, a linear combination of these vectors would be any combination of them using addition and scalar multiplication. A few examples would be:

The vector \(\vec{b} = \left[ \begin{array}{c}3\\ 6\\ 9\end{array} \right]\) is a linear combination of \(\vec{v}_1\), \(\vec{v}_2\), \(\vec{v}_3\).

Why is this true? This vector can be written as a combination of the three given vectors using scalar multiplication and addition. Specifically,

\(\left[ \begin{array}{c}3\\ 6\\ 9\end{array} \right] = 3\left[ \begin{array}{c}1\\ 2\\ 3\end{array} \right] + 0\left[ \begin{array}{c}3\\ 5\\ 1\end{array} \right] + 0\left[ \begin{array}{c}0\\ 0\\ 8\end{array} \right]\)

Or, using the names given to each vector:

\(\vec{b} = 3\vec{v}_1 + 0\vec{v}_2 + 0\vec{v}_3\)

The vector \(\vec{x} = \left[ \begin{array}{c}2\\ 3\\ -6\end{array} \right]\) is a linear combination of \(\vec{v}_1\), \(\vec{v}_2\), \(\vec{v}_3\).

Once again, we can show this is true by showing that you can combine the vectors \(\vec{v}_1\), \(\vec{v}_2\), and \(\vec{v}_3\) using addition and scalar multiplication such that the result is the vector \(\vec{x}\).

\(\left[ \begin{array}{c}2\\ 3\\ -6\end{array} \right] = -1\left[ \begin{array}{c}1\\ 2\\ 3\end{array} \right] + 1\left[ \begin{array}{c}3\\ 5\\ 1\end{array} \right] + \left(-\dfrac{1}{2}\right)\left[ \begin{array}{c}0\\ 0\\ 8\end{array} \right]\)

or again, equivalently

\(\vec{x} = -1\vec{v}_1 +1\vec{v}_2 + \left(-\dfrac{1}{2}\right)\vec{v}_3\)

Of course, we could keep going for a long time as there are a lot of different choices for the scalars and way to combine the three vectors. In general, the set of ALL linear combinations of these three vectors would be referred to as their span. This would be written as \(\textrm{Span}\left(\vec{v}_1, \vec{v}_2, \vec{v}_3\right)\). The two vectors above are elements, or members of this set.

Formal Definition

Now that we have seen a couple of examples and the general idea, let’s finish with the formal definition of a linear combination of vectors. Let the vectors \(\vec{v}_1, \vec{v}_2, \vec{v}_3, \cdots \vec{v}_n\) be vectors in \(\mathbb{R}^{n}\) and \(c_1, c_2, \cdots , c_n\) be scalars. Then the vector \(\vec{b}\), where \(\vec{b} = c_1\vec{v}_1 + c_2\vec{v}_2 + \dots + c_n\vec{v}_n\) is called a linear combination of \(\vec{v}_1, \vec{v}_2, \vec{v}_3, … \vec{v}_n\). The scalars \(c_1, c_2, … , c_n\) are commonly called the “weights”.

Again, this is stating that \(\vec{b}\) is a result of combining the vectors using scalar multiplication (the c’s) and addition.

Study guide – linear combinations and span


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