Take-Home Assignment 3
MAT223 - Winter 2025
1 In this question we let T : R2 → R2 be the linear transformation pictured in the image below.
The image shows you specifically where four input points, 0, a, b, c ∈ R2 are sent by T. Additionally, note that it is implied that: the shaded region on the left is mapped proportionally onto the shaded region on the right; that T(0) = 0, T(a) = a*, T(b) = b* and T(c) = c*; and that the gridlines are spaced at unit distance from each other (i.e. a is the point (1, 2).)
Find the matrix AT (i.e. find the matrix AT so that T(x) = ATx.)
2 Let T : R3 → R3 be projection onto the yz-plane. (That is, each point x ∈ R3
is mapped by T to the closest point to on the yz-plane to x. You may assume this is a linear transformation.)
Find the matrix AT (i.e. the matrix so that T(x) = ATx). Additionally, show that AT has eigenvalue 0 and is diagonalizable.
3 Read the following definition carefully, and then answer the questions below.
Definition A linear transformation T : R2 → R2
is called sticky if it keeps the vectors on one coordinate axis unchanged, while all other vectors are shifted parallel to that axis.
Example For example, the transformation T(x, y) = (x + 3y, y) is sticky, because it does not affect vectors on the x-axis (i.e. T(k, 0) = (k, 0) for all k ∈ R), but vectors off the x-axis are shifted horizontally. That is, if (x, y) is not on the x-axis (i.e. y ≠ 0), then it is mapped by T to a point on the line through (x, y) and (0, y) (which is parallel to the x-axis).
3.1 This will not be marked. Determine the matrix, AT, for the transformation T defined in the Example above, and sketch a copy of R2 with the fundamental parallelogram for T included. Make sure to indicate T(e1) and T(e2) in your picture.
3.2 Show that if R : R2 → R2
is an arbitrary sticky linear transformation that is not the identity transformation, then AR (the matrix for R) is not diagonalizable. Hint: think geometrically about possible eigenvectors of AR.
4 Determine if the statements below are True or False.
If it’s True, explain why. If it’s False explain why not, or give an example demonstrating why it’s false with an explanation. A correct choice of “True” or “False" with no explanation will not receive any credit.
4.1 True or False: If T : R2 → R2
is a linear transformation so that there is exactly one x ∈ R2 with T(x) = x, then T is not invertible (i.e. AT, the matrix for T, is not invertible).
4.2 True or False: If T is a linear transformation which is not invertible (i.e. AT, the matrix of T, is not invertible), then there is some non-zero x in the domain of T so that T(x) = 0.
5 Do not hand this question in! This question explores an interesting application of eigenvectors, but will not be marked.
The Neilsen model is a matrix-based model of population change. The model tracks the changes in a population which is divided into n age groups (for some fixed n ≥ 2.)
The model works by using the size of each age group at the start (the "starting generation") to predict the sizes of the various age groups at some later points (we can say "generation k" for some k). We let x(k) ∈ Rn be the vector whose coordinates are the sizes of the age groups at generation k.
Example For example, if we wrote it would tell us that:
• There are four age groups in the population we are considering (since the vector is in R4
);
• In generation 3 (since it is x
(k)
for k = 3), the size of the first age group is 55, the size of the second age group is 123, etc.
Definition A Neilsen matrix is a matrix of the form.
The Neilsen model works by repeatedly multiplying x(0) (the starting population) by a fixed matrix N of a particular form, called a Neilsen matrix, giving the model the form.
That is, the model predicts the sizes of the age groups in generation k, i.e. x
(k)
, by multiplying the vector x
(k−1)
representing the previous generation by matrix N.
Each step in this iterative process represents the members of age group j moving to age group j + 1, new offspring becoming age group 1, and the members of age group n dying.
The value aj ≥ 0 for j = 1, 2, . . . , n is the expected offspring (new members of age group 1) produced by age group ji. And the number bj
for j = 1, 2, . . . , n is the fraction of individuals from age group j that will survive as they move into age group i + 1.
Theorem If N is a Neilsen matrix, then it will have exactly one positive eigenvalue.
Furthermore, if λ1 is the unique positive eigenvalue of a Neilsen matrix N and λ is any other eigenvalue of N, then |λ| ≤ λ1.
Def For any matrix A, if λ1 is an eigenvalue of A, and |λ| < λ1 for all other eigenvalues λ of A, then λ1 is called the dominant eigenvalue of A.
As stated above, this question will not be marked. If you do it anyway, that’s on you.
Let N be the Neilsen matrix
5.1 This question will not be marked. Determine the eigenvalues of N; then indicate which eigenvalue is the dominant eigenvalue; and finally, find the eigenvector for that eigenvalue.
5.2 This question will not be marked. Suppose that the starting population vector is
Determine the size of age group 3 in generation 50 (i.e., the third coordinate in x(50)) using the same Neilsen matrix N as above.
You may leave any large exponents, like 10500 as-is (you don’t need to use a calculator to simplify them.)