Linear Algebra/Topic: Geometry of Linear Maps/Solutions
- Problem 1
Let be the transformation that rotates vectors clockwise by radians.
- Find the matrix representing with respect to the standard bases. Use Gauss' method to reduce to the identity.
- Translate the row reduction to to a matrix equation (the prior item shows both that is similar to , and that no column operations are needed to derive from ).
- Solve this matrix equation for .
- Sketch the geometric effect matrix, that is, sketch how is expressed as a combination of dilations, flips, skews, and projections (the identity is a trivial projection).
- To represent , recall that rotation counterclockwise by radians is represented with respect to the standard basis in this way.
- The reduction is expressed in matrix multiplication as
- Taking inverses
- Reading the composition from right to left (and ignoring the identity matrices as trivial) gives that has the same effect as first performing this skew
followed by a dilation that multiplies all first components by (this is a "shrink" in that ) and all second components by , followed by another skew.
For instance, the effect of on the unit vector whose angle with the -axis is is this.
Verifying that the resulting vector has unit length and forms an angle of with the -axis is routine.
- Problem 2
What combination of dilations, flips, skews, and projections produces a rotation counterclockwise by radians?
We will first represent the map with a matrix , perform the row operations and, if needed, column operations to reduce it to a partial-identity matrix. We will then translate that into a factorization . Subsitituting into the general matrix
gives this representation.
Gauss' method is routine.
That translates to a matrix equation in this way.
Taking inverses to solve for yields this factorization.
- Problem 3
What combination of dilations, flips, skews, and projections produces the map represented with respect to the standard bases by this matrix?
This Gaussian reduction
gives the reduced echelon form of the matrix. Now the two column operations of taking times the first column and adding it to the second, and then of swapping columns two and three produce this partial identity.
All of that translates into matrix terms as: where
the given matrix factors as .
- Problem 4
Show that any linear transformation of is the map that multiplies by a scalar .
Represent it with respect to the standard bases , then the only entry in the resulting matrix is the scalar .
- Problem 5
Show that for any permutation (that is, reordering) of the numbers , ..., , the map
can be accomplished with a composition of maps, each of which only swaps a single pair of coordinates. Hint: it can be done by induction on . (Remark: in the fourth chapter we will show this and we will also show that the parity of the number of swaps used is determined by . That is, although a particular permutation could be accomplished in two different ways with two different numbers of swaps, either both ways use an even number of swaps, or both use an odd number.)
We can show this by induction on the number of components in the vector. In the base case the only permutation is the trivial one, and the map
is indeed expressible as a composition of swaps— as zero swaps. For the inductive step we assume that the map induced by any permutation of fewer than numbers can be expressed with swaps only, and we consider the map induced by a permutation of numbers.
Consider the number such that . The map
will, when followed by the swap of the -th and -th components, give the map . Now, the inductive hypothesis gives that is achievable as a composition of swaps.
- Problem 6
Show that linear maps preserve the linear structures of a space.
- Show that for any linear map from to , the image of any line is a line. The image may be a degenerate line, that is, a single point.
- Show that the image of any linear surface is a linear surface. This generalizes the result that under a linear map the image of a subspace is a subspace.
- Linear maps preserve other linear ideas. Show that linear maps preserve "betweeness": if the point is between and then the image of is between the image of and the image of .
- A line is a subset of of the form . The image of a point on that line is , and the set of such vectors, as ranges over the reals, is a line (albeit, degenerate if ).
- This is an obvious extension of the prior argument.
- If the point is between the points and then the line from to has in it. That is, there is a such that (where is the endpoint of , etc.). Now, as in the argument of the first item, linearity shows that .
- Problem 7
Use a picture like the one that appears in the discussion of the Chain Rule to answer: if a function has an inverse, what's the relationship between how the function — locally, approximately — dilates space, and how its inverse dilates space (assuming, of course, that it has an inverse)?
The two are inverse. For instance, for a fixed , if (with ) then .