Linear Combination and Span
Using vector addition and scaling together, we can generate new vectors from old ones. Suppose be p vectors in (where n can be 2, 3 and any other natural number). Let be p real numbers. Then we can define a new vector v as follows:
This new vector v is called a linear combination of with weights . In the applet below, you can see the linear combination of vectors u and v in with integer weights (range from -10 to 10). Here we regard vectors as points instead of arrows for better illustration. In the input bar, you can type any such linear combination (e.g. 3u-v) and the arrow pointing to the corresponding point will appear.
Span
Given any vectors in , we consider the set of all linear combinations of them. It is called the set of vectors in spanned (or generated) by . This set is denoted by . In other words, we have
In the applet, you can study for different u and v. Then answer the following questions:
What can you say about u and v when ?
Find out all the possibilities for .
Span in 3D
Given three vectors u, v and w in , we will study the following subsets: and . You can drag the three vectors freely in the applet below.
Given any vectors in . Find out all the possibilities of .
Matrix Representation of Linear Combination
Now we express u, v, and w In as column vectors:
We can write the linear combination of them with weights as follows:
For vectors in another dimensions, you can get a similar column vector for their linear combination. Observe that the arithmetic on column vectors is essentially the arithmetic on the corresponding “entries” on the same row.
Such matrix representation of linear combination is closely related to systems of linear equations, which we will study in detail later.