Recall that to say is a basis for means that spans and is linearly independent. Since spans we can write any in as a linear combination of For uniqueness, suppose that we had two such expressions:
Subtracting the first equation from the second yields
Since is linearly independent, the only solution to the above equation is the trivial solution: all the coefficients must be zero. It follows that for all which proves that
According to the above fact, every vector in can be written as a linear combination of with unique coefficients. For example,
In this case, the coordinates of are exactly the coefficients of
What exactly are coordinates, anyway? One way to think of coordinates is that they give directions for how to get to a certain point from the origin. In the above example, the linear combination can be thought of as the following list of instructions: start at the origin, travel units north, then travel units east, then units down.
Definition
Let be a basis of a subspace and let
be a vector in The coefficients are the coordinates of with respect to . The -coordinate vector of is the vector
If we change the basis, then we can still give instructions for how to get to the point but the instructions will be different. Say for example we take the basis
We can write in this basis as In other words: start at the origin, travel northeast times as far as then units east, then units down. In this situation, we can say that “ is the -coordinate of is the -coordinate of and is the -coordinate of ”
The above definition gives a way of using to label the points of a subspace of dimension a point is simply labeled by its -coordinate vector. For instance, if we choose a basis for a plane, we can label the points of that plane with the points of
These form a basis for a plane in We indicate the coordinate system defined by by drawing lines parallel to the “-axis” and “-axis”:
We can see from the picture that the -coordinate of is equal to as is the -coordinate, so Similarly, we have
Figure8Left: the -coordinates of a vector Right: the vector The violet grid on the right is a picture of the coordinate system defined by the basis one set of lines measures the -coordinate, and the other set measures the -coordinate. Drag the heads of the vectors and to understand the correspondence between and its -coordinate vector.
