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MIT 18.06 Exam 1

The first graded exam: elimination, column and null spaces and the four fundamental subspaces.

1. MIT 18.06 Exam 1 — how this assessment works

This lesson reproduces the Exam “MIT 18.06 Exam 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 4 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 EXAM1 Problem 1 — From the complete solution to $R$, $\mathbf{d}$, $A$, $\mathbf{b}$ (24 pts)

Forward elimination changes $A\mathbf{x}=\mathbf{b}$ to a row reduced $R\mathbf{x}=\mathbf{d}$ : the complete solution is

\mathbf{x}=\begin{bmatrix} 4 \\ 0 \\ 0 \end{bmatrix}+\mathbf{c}_1 \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}+\mathbf{c}_2 \begin{bmatrix} 5 \\ 0 \\ 1 \end{bmatrix}

(a) (14 points) What is the 3 by 3 reduced row echelon matrix $R$ and what is $\mathbf{d}$ ?

(b) (10 points) If the process of elimination subtracted 3 times row 1 from row 2 and then 5 times row 1 from row 3, what matrix connects $R$ and $\mathbf{d}$ to the original $A$ and $\mathbf{b}$ ? Use this matrix to find $A$ and $\mathbf{b}$ .

3. MIT 18.06 EXAM1 Problem 2 — Nullspace, column space, and the rref of the block matrix $\begin{bmatrix} A & A \\ A & A \end{bmatrix}$ (36 pts)

Suppose $A$ is the matrix

A=[ \begin{array}{c c c c} 0 & 1 & 2 & 2 \\ 0 & 3 & 8 & 7 \\ 0 & 0 & 4 & 2 \end{array} ].

(a) (16 points) Find all special solutions to $Ax=0$ and describe in words the whole nullspace of $A$ .

(b) (10 points) Describe the column space of this particular matrix $A$ . “All combinations of the four columns” is not a sufficient answer.

B=[ \begin{array}{c c} A & A \\ A & A \end{array} ] \text{ using the same } A?

4. MIT 18.06 EXAM1 Problem 3 — Consistency and the fundamental subspaces of a full-row-rank $3 \times 5$ matrix (16 pts)

(16 points) Circle the words that correctly complete the following sentence:

(a) Suppose a 3 by 5 matrix $A$ has rank $r=3$ . Then the equation $Ax=b$ (always/sometimes but not always) has (a unique solution/many solutions/no solution).

(b) What is the column space of $A$ ? Describe the nullspace of $A$ .

5. MIT 18.06 EXAM1 Problem 4 — Is a given vector in the column space of $A$? (24 pts)

Suppose that $A$ is the matrix

A=[ \begin{array}{c c} 2 & 1 \\ 6 & 5 \\ 2 & 4 \end{array} ].

(a) (10 points) Explain in words how knowing all solutions to $A\mathbf{x}=\mathbf{b}$ decides if a given vector $\mathbf{b}$ is in the column space of $A$ .

(b) (14 points) Is the vector $\mathbf{b}=[ \begin{array}{c} 8 \\ 28 \\ 14 \end{array} ]$ in the column space of $A$ ?

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