Let V be a vector space and let be a basis for V. Every vector can be uniquely expressed as a linear combination of elements of :
(Let me remind you of why this is true. Since a basis spans, every can be written in this way. On the other hand, if are two ways of writing a given vector, then , and by independence , ..., --- that is, , ..., . So the representation of a vector in this way is unique.)
Consider the situation where is a finite ordered basis --- that is, fix a numbering of the elements of . If , the ordered list of coefficients is uniquely associated with v. The are the components of v with respect to the (ordered) basis ; I will use the notation
It is easy to confuse a vector with the representation of the vector in terms of its components relative to a basis. This confusion arises because representation of vectors which is most familiar is that of a vector as an ordinary n-tuple in :
This amounts to identifying the elements of with their representation relative to the standard basis
Example. (a) Show that
is a basis for .
These are three vectors in , which has dimension 3. Hence, it suffices to check that they're independent. Form the matrix with the elements of as its rows and row reduce:
The vectors are independent. Three independent vectors in must form a basis.
(b) Find the components of relative to .
I must find numbers a, b, and c such that
This is equivalent to the matrix equation
Set up the matrix for the system and row reduce to solve:
This says , , and . Therefore, .
(c) Write in terms of the standard basis.
I'll write for v relative to and for v relative to the standard basis. The matrix equation in (b)
In (b), I knew and I wanted ; this time it's the other way around. So I simply put into the spot and multiply:
Let me generalize the observation I made in (c).
I'll write for M, and call it a translation matrix. Again, translates vectors written in terms of to vectors written in terms of the standard basis.
The inverse of a square matrix M is a matrix such that , where I is the identity matrix. If I multiply the last equation on the left by , I get
In words, this means:
This means that . Dispensing with M, I can say that
In the example above, left multiplication by the following matrix translates vectors from to the standard basis:
The inverse of is
Left multiplication by this matrix translates vectors from the standard basis to .
Example. (Translating vectors from one basis to another) The translation analogy is a useful one, since it makes it easy to see how to set up arbitrary changes of basis.
For example, suppose
is another basis for .
Here's how to translate vectors from to :
Remember that the product is read from right to left! Thus, the composite operation translates a vector to a standard vector, and then translates the resulting standard vector to a vector. Moreover, I have matrices which perform each of the right-hand operations.
This matrix translates vectors from to the standard basis:
This matrix translates vectors from the standard basis to :
Therefore, multiplication by the following matrix will translate vectors from to :
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