Recall:
To give a parametric representation of a linear subspace in a vector space is to represent a general member of the subspace as a linear combination of the vectors in some basis of the subspace.
The coefficients of the basis in such a representation are “coordinates” in the linear subspace relative to the basis.
To have coordinates for the points of a linear subspace of dimension is to have a linear way of matching points in the subspace with points in .
To have coordinates for the points of a linear subspace of dimension is to have an isomorphism from to the subspace.
Proof. The inverse of is an isomorphism from to .
Proof. Compose an isomorphism from to with an isomorphism from to .
Proof. It is an exercise to show that if is a basis of , then is a basis of .
Proof. If is a basis of , then the linear map that is defined by is an isomorphism from to .
Note: When are linearly independent, the coefficients for a given linear combination of them are unique:
Definition. If form a basis of , then are called the coordinates of with respect to when
Example: , , and are the coordinates of the point with respect to the standard basis of .
Example: , , and are the coordinates of the polynomial with respect to the basis of the -dimensional vector space consisting of all polynomials with degree at most in the variable .
The order in which the members of a basis are listed affects the ordering of coordinates taken with respect to that basis.
Recall:
Use and to “transport” to
The transport of is the linear map in this diagram:
is defined by
Since and are isomorphisms, one has
So the theorem is proved by “transport” to the Euclidean case.
Task: If possible, invert the matrix
Form the matrix that augments with the identity matrix , and use row operations to maneuver the first 4 columns of that into reduced row echelon form.
In this case the RREF of the first 4 columns is so the last 4 columns of the reduced matrix form the inverse of , which is:
Task: For the following matrix find
Note: As explained in the previous class, this is essentially the same problem as that of finding linear equations for the image of the linear map
The matrix:
The RREF of its transpose:
The rank of is .
A non-redundant characterizing set of row relations: