代写MAT223H5S - Linear Algebra I - Winter 2025 Make-Up Term Test代做Statistics统计

MAT223H5S - Linear Algebra I - Winter 2025

Make-Up Term Test

1

1.1 (2 points) Suppose that A is a matrix with A −1 = , and b = . Determine the unique solution to the equation Ax = b.

1.2 (3 points) Suppose that T : R2 → R2 first rotates the plane counterclockwise by π/2, then reflects across the line y = x. Determine the matrix AT.

1.3 (6 points) For which values (if any) of c ∈ R is the following set of vectors in R3 linearly independent?

2

Reminder: show your work and justify your steps using only techniques taught in this course.

2.1 (1 point) The equation x − y − 6z = 6 represents a plane in R3 . Determine whether the point P = (3, −3, 0) is on the plane or not.

2.2 (1.5 points) Determine the angle between vectors u = and v =

2.3 (2.5 points) Provide an example, with justification, of a 2 × 2 matrix which is skew-symmetric and invertible, or explain why no such matrix exists.

2.4 (5 points) Find the shortest distance between the lines L1 and L2 with the following equations:

You should include a rough drawing illustrating the situation (e.g. it doesn’t have to plot the lines accurately); the drawing should show any vectors and points that you compute as part of your solution.

3

3.1 (5 points) Show that U is a subspace.

3.2 (5 points) Determine a basis for U. Show your steps.

4

4.1 (5 points) Let T : R3 → R3 be the transformation given by T

Show that T is linear by verifying the two properties below. Do not simply say that T is a matrix transformation, or use any other technique, or you will receive 0 points.

(1) For all u, v ∈ R3 , we have T(u + v) = T(u) + T(v).

(2) For all u ∈ R3 and r ∈ R, we have T(ru) = rT(u).

4.2 (5 points) In the pictures below, the fundamental parallelograms of two linear transformations, S, T : R2 → R2 are shown. Determine all eigenvalues (if any) for each of S and T and for each eigenvalue determine a set of basic eigenvectors for that eigenvalue.

Make sure to justify your answers using geometric arguments only (i.e. ones which reference the pictures of the transformations, not any algebra involving the matrices of the transformations.) Assume the grid lines are spaced 1 unit apart.

5

Determine if the statements below are true or false.

Make sure to justify your answers! You will receive no credit for simply selecting “true" or “false", or providing little explanation.

5.1 True or False: If x, y, z ∈ R3 and {x, y, z} is linearly independent, then {x, x + y + z, z} is also linearly independent.

5.2 True or False: If A is a 3 × 6 matrix, then dim(null(A)) > 2.

5.3 True or False: If A is a 4 × 4 matrix and the systems (A − I)x = 0 and (A + I)x = 0 each have two basic solutions, then A is diagonalizable.

5.4 True or False: If two planes in R3 pass through the origin, and intersect in a line with direction vector d, then d is orthogonal to the normal vectors both of those planes.