A linear combination is a weighted some of other vectors. The following are examples for linear combinations of vectors:

(1)   equation

In general, a vector equation is a linear combination of vectors equation and equation if each can be multiplied by a scalar and the sum is equal to equation: equation for some numbers equation and equation.

Linear combinations and linear independence

Two vectors equation and equation are said to be linear independent, if equation only for equation. This means that vector equation cannot be obtained from vector equation, no matter which number it is multiplied by. Imagine a two-dimensional coordinate system and you can imagine the x-axis is described by a vector equation and the y-axis by a vector equation. Any vector that we can obtain by multiplying equation and equation by some number and then adding them is linear dependent on them. Let’s say, we want to add another dimension, the z-axis. If the vector equation for the z-axis is a linear combination of equation and equation (it is linear dependent), then any linear combination of all equation, equation and equation will have a zero for the z-component. For example let the vectors be defined as follows:

(2)   equation

If equation is a linear combination of the other vectors, then equation, and no matter which numbers we use for equation and equation, the z-component will always be zero, because it is zero in both equation and equation. To make equation linear independent of equation and equation, let us define equation as follows:

    equation

Now using a linear combination of all three vectors can give us a new vector that points at any arbitrary coordinate in the coordinate system formed by these three vectors. The new vector equation is no longer a linear combination of equation and equation and thus, it is linear independent of them.

Given two vectors equation and equation, we want to determine whether they are linear dependent or independent. In order to do this, we substitute them into the equation shown above:

(3)   equation

To solve this, we create this system of linear equations:

    equation

We multiply the first row by 3 and then subtract the second row from it:

    equation

We find that equation and substituting this into the second equation and solving for equation gives equation. Thus, the two vectors are linear independent. Let us define a third vector equation and check if it is linear dependent on the others.

(4)   equation

Obviously, zero is one solution but we want to see if there is a solution where at least one of the equation values is non-zero, in which case there would be a linear dependency. Let us create the system of linear equations

    equation

Multiplying the first equation by 3 and adding the second one will give:

    equation

From the first equation we know that equation and if we substitute this into the second equation and solve it for equation, we get:

    equation

So the solution is equation, for example equation solve the system of equations, thus we have shown that these three vectors are linear dependent because there is a solution where not all of the equation values are zero.

For more advanced reading on linear combinations of vectors, have a look at the Wikipedia article at http://en.wikipedia.org/wiki/Linear_combination.