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
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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.
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The unique matrix is
For a given real number find a matrix for which the linear function defined by is the counterclockwise rotation of the plane through the angle of (radian) measure .
“ahead” of by
Find a matrix for which the linear function given by is the reflection of in the plane (where the coordinate ).
Points in the plane do not move.
Points on the -axis are “flipped”, i.e.,
When is an matrix, the phrase “corresponding linear function” will denote the linear function defined by In the case compute each of the following items both for (i) itself and for (ii) its reduced row echelon form:
The reduced row echelon form is
Let be the -dimensional vector space consisting of all polynomials of degree or less, and let be the familiar basis of Let be the linear map that is defined by where and denote the first and second derivatives of Find the matrix of with respect to the basis , i.e., find the matrix that appears in the transport diagram
Compute the polynomials for .
The columns of are the coefficient vectors for these polynomials.