Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due

Take-Home Assignment 4

MAT223 - Winter 2025

1 Let 

1.1 Find a basis B1 for im(A), and a basis B2 for null(A).

1.2 Explain briefly why im(A) is a plane and find the equation of this plane.

1.3 Explain briefly why null(A) is a line, and find the equation of this line.

1.4 Find a basis B3 for im(A T ) and a basis B4 for null(A T ).

1.5 Show that B1 ∪ B4 and B2 ∪ B3 are both bases for R3 (!)

Note: for sets X and Y, their union, written X ∪ Y, is defined to be {x | x ∈ X or x ∈ Y}.

2 Read Section 5.1, Example 5.1.3 (on page 263) and the paragraphs immediately before and immediately after it, in our Course Textbook, https://q.utoronto.ca/courses/378311/files/35471615, then answer the following questions:

2.1 Determine the non-trivial eigenspaces of the matrix  and find a basis for each. What is the dimension of each non-trivial eigenspace? What is the sum of their dimensions, and how does this number relate to the diagonalizability (or non-diagonalizability) of A?

2.2 Suppose that A and B are 3 × 3 matrices so that:

• E1(A) is the xy-plane, and E0(A) is the z-axis.

• E−1(B) is the xy-plane, and E2(B) is the z-axis.

Show that AB = BA.

3 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.

3.1 True or False: If A is an n × n matrix and dim(null(A − I)) = n, then A is invertible. Hint: think about eigenspaces (see above).

3.2 True or False: If 0 < θ < 2π, then Rθ , the linear transformation R2 → R2 that rotates counterclockwise by θ, has only trivial eigenspaces.

3.3 True or False: If T and S are linear transformations R2 → R2 , and {u, v} is a linearly independent set in R2 , and T(u) = S(u) and T(v) = S(v), then T(x) = S(x) for all x ∈ R2 .