MAT223H5S - Linear Algebra I - Winter 2025
Term Test 2 - Version A
1
1.1 (2 points) Let d = and a = . Determine whether or not d and a are orthogonal.
1.2 (3 points) Compute ||projd
(a)||. Use d and a as above. You do not need to simplify fractions, roots, etc.
1.3 (1 point) Let T be a linear transformation with matrix AT = . What is the domain of T?
(Is it R2
, R3
, or something else?) You do not need to justify your answer.
1.4 (4 points) Let A = Is d in im(A)? (Use d as on the previous page.)
2
2.1 (5 points) Find the equation of the plane through the points
P = (1, 2, 3), Q = (−1, 9, −2), R = (1, 0, 4).
Give your answer in the form. ax + by + cz = d.
You should only use the methods taught to you in this course to solve this problem. In particular, do not apply any formulas or shortcuts from outside the course.
2.2 (5 points) Show that the function F : R2 → R2 given by is not a linear transformation.
3
3.1 (4 points)
Show that the set U defined below is a subspace. As part of your answer, find a spanning set for U.
3.2 (6 points)
Let
For each of the three subspace axioms below, determine whether the set V satisfies it or not.
In particular, if you think that V satisfies a given axiom, explain why. If you think that V does not satisfy a given axiom, give an example showing that.
(1) 0 ∈ V.
(2) For all u, v ∈ V, we have u + v ∈ V.
(3) For all v ∈ V and t ∈ R, we have tv ∈ V.
4
Let T : R2 → R2 be the linear transformation which reflects across the line y = −2x and then stretches in the x and y directions, each by a factor of 5.
You may assume that
4.1 (5 points) Sketch a single copy of R2 which contains the following:
• e1, e2, T(e1) and T(e2),
• The fundamental parallelogram for T (i.e. the image of the unit square under T),
• The lines y = −2x and y = 2/1 x.
Note: For the questions on this page, you must justify your answers using only geometric explanations explicitly relying on and referring to your drawing from the previous part, or to a new drawing. Algebraic work (e.g. computations involving AT) can be used to check your work, but will not be marked.
4.2 (2.5 points) Explain geometrically why your drawing from the previous part implies that neither e1 nor e2 is an eigenvector for AT.
4.3 (2.5 points) Determine two basic eigenvectors v1 and v2 for AT geometrically using your drawing from the first part of this question
(You do not need to determine the eigenvalue(s) associated to those basic eigenvectors.)
5
(2.5 points each = 10 points)
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: Let A be an n × n matrix, and b ∈ Rn be arbitrary. Then the set of solutions to the system Ax = b is a subspace.
5.2 True or False: If a, b, c ∈ R3 and span{a, b, c} is a plane, then span{a, b} is also plane.
5.3 True or False: The system of equations below represents three planes in R3
that intersect in a line:
5.4 True or False: Suppose that U is a subset of R3
such that is in U, but and are not in U. Then U is not a subspace of R3
.