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MAT 223F: Linear Algebra

Problem Set 5.

Instructions

• You need to submit this assignment electronically. If you write your solutions by hand, you need to scan them. Submissions will be made to Crowdmark.

• You will need to submit the answer to each question separately.

• This problem set is about the matrix expression of a linear transformation in an arbitrary basis, finding inverses of matrices and determinants.

• Your solutions will be graded based on both the mathematical ideas involved and your ability to rigorously communicate them. Make sure you use precise language and proper proof writing format. All your solutions must be written using complete English sentences and your arguments should be presented in a cohesive manner.   In particular,  solutions that do not present any justification will not be granted any marks.

• Due to time limitations, we may not grade every question on the assignment.

In this Problem Set, you will work on how to express lines and planes in different ways, as well as develop some geometric intuition relating lines and planes.

1. Consider the system of linear equations

          x                     −z     = −2

(a) Let M be the matrix from the matrix equation in the previous point. Find rref(M).

(b) Consider the matrix X of size 3 × 6 given by X = [M|I] where I is the identity matrix of size 3 × 3. If P is any matrix, express the matrix product PX in terms of M and I .

(c) Find the elementary matrices E1 , . . . ,Er  that take M to its rref. Use parts a) and b) to determine if M is invertible, and if so, find a formula for M − 1 .

(d) Use part c) to find the complete solution set of the system of equations.

2. Let β = {v⃗1 , . . . ,  ⃗vn} ∈ Rn be a basis of Rn . Suppose T : Rn  → Rm is a linear transformation such that {T(v1 ), . . . ,T(vn)} is a linearly independent set. Show that n(T) = 0.

3. Consider the basis β = {e⃗1 ,  ⃗e1 +e⃗2 ,  ⃗e1 +e⃗2 +e⃗3 } in R3 . Find the matrix of the projection onto the y-axis in the basis β .

4. Suppose A is a 4 × 5 matrix such that the linear transformation induced by A, TA  : R5 → R4 defined by TA(x) = Ax is surjective. Find the dimension of the row space of A.

5. Use row reduction and elementary matrices to compute the determinant of the matrix

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