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

This lesson reproduces the Problem Set “MIT 18.06 Problem Set 1” 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 13 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 PSET1 Problem 1 — Cosine of the angle between two vectors from their dot product (4 pts)

The figure shows that $\cos(\alpha)=v_1/\|v\|$ and $\sin(\alpha)=v_2/\|v\|$ . Similarly $\cos(\beta)$ is $\underline{\hspace{2cm}}$ and $\sin(\beta)$ is $\underline{\hspace{2cm}}$ . The angle $\theta$ is $\beta-\alpha$ . Substitute into the trigonometry formula $\cos(\alpha)\cos(\beta)+\sin(\beta)\sin(\alpha)$ for $\cos(\beta-\alpha)$ to find $\cos(\theta)=v \cdot w/\|v\|\|w\|$ .

3. MIT 18.06 PSET1 Problem 2 — Three vectors in the plane with pairwise negative dot products (12 pts)

Can three vectors in the $xy$ plane have $u \cdot v < 0$ and $v \cdot w < 0$ and $u \cdot w < 0$ ?

4. MIT 18.06 PSET1 Problem 3 — A dependent combination of three column vectors (4 pts)

Find a combination $x_1w_1+x_2w_2+x_3w_3$ that gives the zero vector:

w_1=\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}; \quad w_2=\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}; \quad w_3=\begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix}.

Those vectors are (independent)(dependent). The three vectors lie in a $\underline{\hspace{2cm}}$ .

The matrix $W$ with those columns is not invertible.

5. MIT 18.06 PSET1 Problem 4 — The singular 5 by 5 centered difference matrix (12 pts)

The very last words say that the 5 by 5 centered difference matrix is not invertible. Write down the 5 equations $Cx=b$ . Find a combination of left sides that gives zero. What combination of $b_1, b_2, b_3, b_4, b_5$ must be zero?

6. MIT 18.06 PSET1 Problem 5 — Iterating a Markov matrix from $u_0 = (1, 0)$ (4 pts)

Start with the vector $u_0=(1, 0)$ . Multiply again and again by the same "Markov matrix" $A=[.8 \ .3; .2 \ .7]$ . The next three vectors are $u_1, u_2, u_3$ :

u_1=\begin{bmatrix} .8 & .3 \\ .2 & .7 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix}=\begin{bmatrix} .8 \\ .2 \end{bmatrix} \quad u_2=Au_1=\_\_\_\_\_\_ \quad u_3=Au_2=\_\_\_\_\_\_.

What property do you notice for all four vectors $u_0, u_1, u_2, u_3$ .

7. MIT 18.06 PSET1 Problem 6 — Convergence of the Markov chain to its steady state vector (12 pts)

Continue Problem 29 from $u_0=(1, 0)$ to $u_7$ , and also from $v_0=(0, 1)$ to $v_7$ . What do you notice about $u_7$ and $v_7$ ? Here are two MATLAB codes, with while and for. They plot $u_0$ to $u_7$ and $v_0$ to $v_7$ .

The $u$ 's and the $v$ 's are approaching a steady state vector $s$ . Guess that vector and check that $As=s$ . If you start with $s$ , then you stay with $s$ .

8. MIT 18.06 PSET1 Problem 7 — A third equation that makes a plane system singular (4 pts)

Three planes can fail to have an intersection point, even if no planes are parallel. The system is singular if row 3 of $A$ is a $\underline{\hspace{2cm}}$ of the first two rows. Find a third equation that can't be solved together with $x+y+z=0$ and $x-2y-z=1$ .

9. MIT 18.06 PSET1 Problem 8 — Singular systems: rows, columns, and a 100 by 100 example (12 pts)

Start with 100 equations $Ax=0$ for 100 unknowns $x=(x_1, \ldots, x_{100})$ . Suppose elimination reduces the 100th equation to $0=0$ , so the system is "singular".

(a) Elimination takes linear combinations of the rows. So this singular system has the singular property: Some linear combination of the 100 rows is $\underline{\hspace{2cm}}$ .
(b) Singular systems $Ax=0$ have infinitely many solutions. This means that some linear combination of the 100 columns is $\underline{\hspace{2cm}}$ .
(c) Invent a 100 by 100 singular matrix with no zero entries.
(d) For your matrix, describe in words the row picture and the column picture of $Ax=0$ . Not necessary to draw 100-dimensional space.

10. MIT 18.06 PSET1 Problem 9 — What a row operation does to the components of $Ax$ (4 pts)

The entries of $A$ and $x$ are $a_{ij}$ and $x_j$ . So the first component of $Ax$ is $\sum a_{1j}x_j=a_{11}x_1+\cdots+a_{1n}x_n$ . If $E_{21}$ subtracts row 1 from row 2, write a formula for

(a) the third component of $Ax$ (b) the (2,1) entry of $E_{21}A$ (c) the (2,1) entry of $E_{21}(E_{21}A)$ (d) the first component of $E_{21}Ax$ .

11. MIT 18.06 PSET1 Problem 10 — Reducing Pascal's matrix to $I$ by elimination (12 pts)

Find the triangular matrix $E$ that reduces "Pascal's matrix" to a smaller Pascal:

E \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 1 & 3 & 3 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 \end{bmatrix}.

Which matrix $M$ (multiplying several $E$ 's) reduces Pascal all the way to $I$ ?

12. MIT 18.06 PSET1 Problem 11 — Solving $Ax = b$ for the unit right sides (4 pts)

Suppose you solve $Ax=b$ for three special right sides $b$ :

Ax_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}; \quad Ax_2=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}; \quad Ax_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}.

If the three solutions $x_1$ , $x_2$ , $x_3$ are the columns of a matrix $X$ , what is $A$ times $X$ ?

13. MIT 18.06 PSET1 Problem 12 — Where to put the parentheses in a matrix product (12 pts)

Suppose $A$ is $m$ by $n$ , $B$ is $n$ by $p$ , and $C$ is $p$ by $q$ . Then the multiplication count for $(AB)C$ is $mnp+mpq$ . The multiplication count from $A$ times $BC$ with $mnq+npq$ separate multiplications.

(a) If $A$ is 2 by 4, $B$ is 4 by 7, and $C$ is 7 by 10, do you prefer $(AB)C$ or $A(BC)$ ?
(b) With $N$ -component vectors, would you choose $(u^T v)w^T$ or $u^T(vw^T)$ ?
(c) Divide by $mnpq$ to show that $(AB)C$ is faster when $n^{-1}+q^{-1} < m^{-1}+p^{-1}$ .

14. MIT 18.06 PSET1 Problem 13 — Row $1 +$ row $2 =$ row $3$: why $A$ is not invertible (4 pts)

If $A$ has row $1+$ row $2=$ row $3$ , show that $A$ is not invertible:

(a) Explain why $Ax=(1, 0, 0)$ cannot have a solution.
(b) Which right sides $(b_1, b_2, b_3)$ might allow a solution to $Ax=b$ ?
(c) What happens to row 3 in elimination?