1. MIT 18.06 Problem Set 4 — how this assessment works

This lesson reproduces the Problem Set “MIT 18.06 Problem Set 4” from MIT OpenCourseWare 18.06 Linear Algebra (Spring 2010) (https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/), licensed under CC BY-NC-SA 4.0. Work through 10 problems in order: type your reasoning in the answer box or write it on the handwriting pad, then submit for AI grading.

How the grading works Every problem is graded by an AI mathematics teacher that reads the official solution shipped with the course. It checks your work criterion by criterion, awards partial credit for correct steps, and returns written feedback. Equivalent mathematics and valid alternative methods are accepted — a different route to the same result can still earn full marks.

If your answer is incomplete, the feedback names exactly which steps are missing, so you can correct them and try again.

2. MIT 18.06 PSET4 Problem 1 — Largest independent set among six vectors (4 pts)

(Recommended) Find the largest possible number of independent vectors among

v_1=\begin{bmatrix} 1 \\ -1 \\ 0 \\ 0 \end{bmatrix} \quad v_2=\begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 \end{bmatrix} \quad v_3=\begin{bmatrix} 1 \\ 0 \\ 0 \\ -1 \end{bmatrix} \quad v_4=\begin{bmatrix} 0 \\ 1 \\ -1 \\ 0 \end{bmatrix} \quad v_5=\begin{bmatrix} 0 \\ 1 \\ 0 \\ -1 \end{bmatrix} \quad v_6=\begin{bmatrix} 0 \\ 0 \\ 1 \\ -1 \end{bmatrix}.

3. MIT 18.06 PSET4 Problem 2 — Basis of a plane, its trace on the xy-plane, and its normals (4 pts)

Find a basis for the plane $x-2y+3z=0$ in $\mathbb{R}^3$ . Then find a basis for the intersection of that plane with the $xy$ plane. Then find a basis for all vectors perpendicular to the plane.

4. MIT 18.06 PSET4 Problem 3 — Matrices that commute with the shift matrix (4 pts)

If $AS=SA$ for the shift matrix $S$ , show that $A$ must have this special form:

If

[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}] [\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}]=[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}] [\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}], \text{ then } A=[\begin{array}{lll} a & b & c \\ 0 & a & b \\ 0 & 0 & ai \end{array}].

"The subspace of matrices that commute with the shift $S$ has dimension $\underline{\hspace{2cm}}$ ."

5. MIT 18.06 PSET4 Problem 4 — The identity from the five nontrivial permutation matrices (12 pts)

Write a 3 by 3 identity matrix as a combination of the other five permutation matrices. Then show that those five matrices are linearly independent. (Assume a combination gives $c_1P_1+\cdots+c_5P_5=0$ , and check entries to prove $c_i$ is zero.) The five permutation matrices are a basis for the subspace of 3 by 3 matrices with row and column sums all equal.

6. MIT 18.06 PSET4 Problem 5 — Dimensions of outputs, nullspace and inputs (12 pts)

(An aside in the text, followed by) \emph{dimension of outputs+dimension of nullspace=dimension of inputs}. For an $m$ by $n$ matrix of rank $r$ , what are those 3 dimensions? Outputs=column space. This question will be answered in Section 3.6, can you do it now?

7. MIT 18.06 PSET4 Problem 6 — Rank inequalities when Ax = b is sometimes unsolvable (4 pts)

$A$ is an $m$ by $n$ matrix of rank $r$ . Suppose there are right sides $\mathbf{b}$ for which $A\mathbf{x}=\mathbf{b}$ has no solution.

(a) What are all the inequalities ( $<$ or $\leq$ ) that must be true between $m$ , $n$ , and $r$ ?

(b) How do you know that $A^T\mathbf{y}=\mathbf{0}$ has solutions other than $\mathbf{y}=\mathbf{0}$ ?

8. MIT 18.06 PSET4 Problem 7 — Solvability and uniqueness of A^T y = d (4 pts)

$A^T\mathbf{y}=\mathbf{d}$ is solvable when $\mathbf{d}$ is in which of the four subspaces? The solution is unique when the $\underline{\hspace{2cm}}$ contains only the zero vector.

9. MIT 18.06 PSET4 Problem 8 — Checkerboard and chess matrices: ranks and bases (4 pts + 12 pts (challenge))

Find the ranks of the 8 by 8 checkerboard matrix $B$ and the chess matrix $C$ :

B=[\begin{array}{cccccccc} 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \end{array}] \text{ and } C=[\begin{array}{cccccccc} r & n & b & q & k & b & n & r \\ p & p & p & p & p & p & p & p \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ p & p & p & p & p & p & p & p \\ r & n & b & q & k & b & n & r \end{array}]

The numbers $r, n, b, k, q, p$ are all different. Find bases for the rowspace and left nullspace of $B$ and $C$ . Challenge problem: Find a basis for the nullspace of $C$ .

10. MIT 18.06 PSET4 Problem 9 — Four fundamental subspaces of a rank-1 2x2 matrix (12 pts)

If $A=\mathbf{u}\mathbf{v}^T$ is a 2 by 2 matrix of rank 1, redraw Figure 3.5 to show clearly the Four Fundamental Subspaces. If $B$ produces those same four subspaces, what is the exact relation of $B$ to $A$ ?

11. MIT 18.06 PSET4 Problem 10 — Left multiplication as a linear map on 3x3 matrices (12 pts)

$M$ is the space of 3 by 3 matrices. Multiply each matrix $X$ in $M$ by

A=\begin{bmatrix} 1 & 0 &-1 \\ -1 & 1 & 0 \\ 0 &-1 & 1 \end{bmatrix}.

Notice: $A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ .

(a) Which matrices $X$ lead to $AX=0$ ?

(b) Which matrices have the form $AX$ for some matrix $X$ ?

(a) finds the "nullspace" of that operation $AX$ and (b) finds the "column space". What are the dimensions of those two subspaces of $M$ ? Why do the dimensions add to $(n-r)+r=9$ ?