1. MIT 18.06 Problem Set 6 — how this assessment works

This lesson reproduces the Problem Set “MIT 18.06 Problem Set 6” 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 PSET6 Problem 1 — Writing the least-squares error as four squares (4 pts)

Write down $E=\|Ax-b\|^2$ as a sum of four squares—the last one is $(C+4D-20)^2$ . Find the derivative equations $\partial E/\partial C=0$ and $\partial E/\partial D=0$ . Divide by 2 to obtain the normal equations $A^T A \widehat{x}=A^T b$ .

3. MIT 18.06 PSET6 Problem 2 — Closest line through the origin (4 pts)

Find the closest line $b=Dt$ , through the origin, to the same four points. An exact fit would solve $D \cdot 0=0$ , $D \cdot 1=8$ , $D \cdot 3=8$ , $D \cdot 4=20$ . Find the 4 by 1 matrix $A$ and solve $A^T A \widehat{x}=A^T b$ . Redraw figure 4.9a showing the best line $b=Dt$ and the e's.

4. MIT 18.06 PSET6 Problem 3 — Closest parabola and the normal equations (4 pts)

Form the closest parabola $b=C+Dt+Et^2$ to the same four points, and write down the unsolvable equations $Ax=b$ in three unknowns $x=(C, D, E)$ . Set up the three normal equations $A^T A \widehat{x}=A^T b$ (solution not required). In figure 4.9a you are now fitting a parabola to 4 points—what is happening in Figure 4.9b?

5. MIT 18.06 PSET6 Problem 4 — Best-fit plane at the corners of a square (12 pts)

Find the plane that gives the best fit to the 4 values $b=(0, 1, 3, 4)$ at the corners $(1, 0)$ and $(0, 1)$ and $(-1, 0)$ and $(0,-1)$ of a square. The equations $C+Dx+Ey=b$ at those 4 points are $Ax=b$ with 3 unknowns $x=(C, D, E)$ . What is $A$ ? At the center $(0, 0)$ of the square, show that $C+Dx+Ey$ is the average of the $b$ 's.

6. MIT 18.06 PSET6 Problem 5 — Unique hyperplane through n points (12 pts)

Usually there will be exactly one hyperplane in $\mathbb{R}^n$ that contains the $n$ given points $x=0, a_1, \ldots, a_{n-1}$ . (Example for n=3: There will be exactly one plane containing $0, a_1, a_2$ unless $\underline{\hspace{2cm}}$ .) What is the test to have exactly one hyperplane in $\mathbb{R}^n$ ?

7. MIT 18.06 PSET6 Problem 6 — Orthonormal vectors are linearly independent (4 pts)

Orthonormal vectors are automatically linearly independent.

(a) Vector proof: When $c_1 q_1+c_2 q_2+c_3 q_3=0$ , what dot product leads to $c_1=0$ ? Similarly $c_2=0$ and $c_3=0$ . Thus, the $q$ 's are independent.

(b) Matrix proof: Show that $Qx=0$ leads to $x=0$ . Since $Q$ may be rectangular, you can use $Q^T$ but not $Q^{-1}$ .

8. MIT 18.06 PSET6 Problem 7 — Gram-Schmidt on three 4-vectors (4 pts)

Find the orthonormal vectors $A, B, C$ by Gram-Schmidt from $a, b, c$ :

a=(1,-1, 0, 0) \qquad b=(0, 1,-1, 0) \qquad c=(0, 0, 1,-1).

Show $\{A, B, C\}$ and $\{a, b, c\}$ are bases for the space of vectors perpendicular to $d=(1, 1, 1, 1)$ .

9. MIT 18.06 PSET6 Problem 8 — QR factorization of a bidiagonal matrix (12 pts)

Factor $[Q, R]=\mathbf{qr}(\mathbf{A})$ for $A=\mathbf{eye}(4)-\mathbf{diag}([111],-1)$ . You are orthogonalizing the columns $(1,-1, 0, 0)$ , $(0, 1,-1, 0)$ , $(0, 0, 1,-1)$ , and $(0, 0, 0, 1)$ of $A$ . Can you scale the orthogonal columns of $Q$ to get nice integer components?

10. MIT 18.06 PSET6 Problem 9 — Subspaces spanned by Q1 and Q2 (12 pts)

If $A$ is $m$ by $n$ , $\mathbf{qr}(\mathbf{A})$ produces a \emph{square} $A$ and zeroes below $R$ : The factors from MATLAB are ( $m$ by $m$ ) ( $m$ by $n$ )

A=\begin{bmatrix} Q_1 & Q_2 \end{bmatrix} \begin{bmatrix} R \\ 0 \end{bmatrix}.

The $n$ columns of $Q_1$ are an orthonormal basis for which fundamental subspace? The $m-n$ columns of $Q_2$ are an orthonormal basis for which fundamental subspace?

11. MIT 18.06 PSET6 Problem 10 — Row sums and the determinant (4 pts)

If the entries in every row of $A$ add to zero, solve $Ax=0$ to prove $\det A=0$ . If those entries add to one, show that $\det(A-I)=0$ . Does this mean $\det A=I$ ?

12. MIT 18.06 PSET6 Problem 11 — The 3x3 Vandermonde determinant (4 pts)

Use row operations to show that the 3 by 3 "Vandermonde determinant" is

\det \begin{bmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{bmatrix}=(b-a)(c-a)(c-b).

13. MIT 18.06 PSET6 Problem 12 — Hilbert matrix determinants and pivots (12 pts)

(MATLAB) The Hilbert matrix $\mathbf{hilb}(n)$ has $i, j$ entry equal to $1/(i+j-1)$ . Print the determinants of $\mathbf{hilb}(1), \mathbf{hilb}(2), \ldots, \mathbf{hilb}(10)$ . Hilbert matrices are hard to work with! What are the pivots of $\mathbf{hilb}(5)$ ?

14. MIT 18.06 PSET6 Problem 13 — Typical determinants of random matrices (12 pts)

(MATLAB) What is the typical determinant (experimentally) of $\mathbf{rand}(n)$ and $\mathbf{randn}(n)$ for $n=50, 100, 200, 400$ ? (And what does "Inf" mean in MATLAB?)