In this section, we interpret a basis of a subspace as a coordinate system on and we learn how to write a vector in in that coordinate system.
If is a basis for a subspace then any vector in can be written as a linear combination
in exactly one way.
When we combine this observation with the pivotal theorem in Section 3.2, we deduce the uniqueness of the reduced row echelon form, already stated in this theorem in Section 2.2. Indeed, each non-pivot column in the reduced row echelon form consists of the uniquely determined coefficients expressing it as a linear combination of the pivot columns to its left, followed by zeros.
Consider the standard basis of from this example in Section 3.4:
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.
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
Let
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
If is a basis for a subspace and is in then
Finding the -coordinates of means solving the vector equation
in the unknowns This generally means row reducing the augmented matrix
