Lecture 10
Recall
Previously, we saw the inverse of a matrix :
where is an matrix; that is, must be square to have an inverse.
In a matrix, this equates to solving each of
which can be formulated as the process of transforming an extended augmented matrix through row reductions:
Fundamental Theorem of Invertible Matrices
Proof of (6).
Consider . If has an inverse , then
Proof of (7).
Suppose is an arbitrary vector. Next consider the system
so the system has a unique solution for any .
Let us take a closer look at some of the implications of this theorem. Suppose is a matrix. Suppose each of the following systems
has a solution, called , respectively.
Suppose we want to find the solution of . Note that
So the solution is given by .
Elementary Matrices
An elementary matrix, , is a matrix we obtain by performing exactly one row operation on an identity matrix.
For example, in the case,
The matrices are all elementary matrices.
Consider multiplying a matrix by the second of these elementary matrices:
We find that multiplying by an elementary matrix has the same effect as performing the same single row operation on as was performed to obtain the elementary matrix. This shows us that to perform one row operation on a matrix we just need to multiply on the left by an appropriate elementary matrix.
Properties of Inverses
For a matrix , the following are true of its inverse :
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1.
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2.
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3.
is unique
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4.
Proof.
Suppose and act as inverses of . That is,
Note then that
So . β
Proof.
Assume that and are both invertible.
Similarly we can show that . β
Question:
Can be invertible while one of or is not invertible? Must one be invertible, must both be invertible?
What if is not invertible? By the fundamental theorem of invertible matrices, we have that must have a non-trivial solution, . Then
Then the equation also has a non-trivial solution. Thus cannot be invertible.
Elementary Matrices and Invertible Matrices
Consider the following matrix and its row reduced echelon form:
By the fundamental theorem of matrices, this shows us that is invertible.
Note though that we have said that we can represent row operations as elementary matrices. Corresponding to these row operations then, are the matrices
If we call the matrices between and as and , then we have that
and so is the inverse of :
so
but this assumes that the matrices are invertible. This brings us to an important question:
Question:
Are elementary matrices invertible? In essence, this equates to asking how to reverse a row operation. Every row operation can be reversed, and so every elementary matrix is invertible.
Vector Space
The term vector space shouldnβt be confused for what we know as vectors. Consider the set
That is, elements of are of the form
where .
Let . It is apparent then that and for any , .
This is very similar to . Consider . Then and for any , .
In fact, many things besides standard vectors will be considered as vectors. Polynomials, functions, and matrices can all be seen as vectors in their respective vector spaces.