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:
Question. What is the matrix for the rotation counterclockwise by the angle with measure (radians)?
Note. A rotation of about the origin is a linear map because, as a rigid motion of the plane, it carries parallelograms to parallelograms, and addition of points in follows the “parallelogram law”.
Observations.
If is the matrix of , then
is the matrix of with respect to the basis pair
Question. If is the plane in that is the linear span of the vectors and is the rotation in space through about the axis through the origin that is perpendicular to , specify a basis of relative to which the matrix of is relatively simple.
Answer. There is slight ambiguity since it is not possible to distinguish between clockwise and counterclockwise.
is a basis of the plane
One computes the “dot product”:
So and are perpendicular.
One of the two possible rotations through will satisfy:
The “cross product” lies on the axis of rotation:
Take , a vector on the axis, as a third basis vector for .
With as selected pairs of bases, the matrix of is:
We have:
Note: The second matrix is the matrix of the linear map with respect to the basis pair .
The first matrix is not the matrix of a linear map but, rather, a matrix whose columns are the standard coordinates — coordinates with respect to the standard basis — of the members of the basis .
The matrix corresponding in a similar way to the standard basis is the identity matrix, and it would be more precise, instead of writing to use to relate the row of vectors to the row of vectors : is the matrix for change of basis between the basis and the standard basis .
For the standard matrix of one has or
Since is linear, and , one has and, therefore, combining the various formulas: yielding the following relation among ordinary matrices:
Because this particular matrix consists of mutually perpendicular columns, all of the same length, it is particularly easy to invert:
is the standard matrix of one of the two rotations through the angle about the line through the origin and the point .
Let be the linear map from to that is defined by where is the matrix Find a matrix for which the linear map given by multiplication by has the property that both and for all and all in .
is the inverse map to . It is the linear map given by the inverse matrix.
The inverse matrix:
Let be a linear map from to for which
.
.
.
Find all possible matrices for which the formula is valid for all in .
Hint: Use the rules for abstract linearity to work out what happens under to and .
is determined by its values on the members of a basis.
is a set of linearly independent vectors in , hence, a basis of .
The columns of are the values of on the standard basis.
.
.
.
The unique matrix is