A linear combination is a weighted some of other vectors. The following are examples for linear combinations of vectors:
In general, a vector is a linear combination of vectors and if each can be multiplied by a scalar and the sum is equal to : for some numbers and .
Linear combinations and linear independence
Two vectors and are said to be linear independent, if only for . This means that vector cannot be obtained from vector , 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 and the y-axis by a vector . Any vector that we can obtain by multiplying and 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 for the z-axis is a linear combination of and (it is linear dependent), then any linear combination of all , and will have a zero for the z-component. For example let the vectors be defined as follows:
If is a linear combination of the other vectors, then , and no matter which numbers we use for and , the z-component will always be zero, because it is zero in both and . To make linear independent of and , let us define as follows:
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 is no longer a linear combination of and and thus, it is linear independent of them.
Given two vectors and , we want to determine whether they are linear dependent or independent. In order to do this, we substitute them into the equation shown above:
To solve this, we create this system of linear equations:
We multiply the first row by 3 and then subtract the second row from it:
We find that and substituting this into the second equation and solving for gives . Thus, the two vectors are linear independent. Let us define a third vector and check if it is linear dependent on the others.
Obviously, zero is one solution but we want to see if there is a solution where at least one of the values is non-zero, in which case there would be a linear dependency. Let us create the system of linear equations
Multiplying the first equation by 3 and adding the second one will give:
From the first equation we know that and if we substitute this into the second equation and solve it for , we get:
So the solution is , for example 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 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.