MIT 18.06 Exam 2

This lesson reproduces the Exam “MIT 18.06 Exam 2” 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 3 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.

(30 points)

(a) Find the matrix $P$ that projects every vector $b$ in $R^3$ onto the line in the direction of $a=(2, 1, 3)$.

(b) What are the column space and nullspace of $P$? Describe them geometrically and also give a basis for each space.

(c) What are \emph{all} the eigenvectors of $P$ and their corresponding eigenvalues? (You can use the geometry of projections, not a messy calculation.) The diagonal entries of $P$ add up to $\underline{\hspace{2cm}}$.

(30 points)

(a) $p=A\widehat{x}$ is the vector in $C(A)$ nearest to a given vector $b$. If $A$ has independent columns, what equation determines $\widehat{x}$? What are all the vectors perpendicular to the error $e=b-A\widehat{x}$? What goes wrong if the columns of $A$ are dependent?

(b) Suppose $A=QR$ where $Q$ has orthonormal columns and $R$ is upper triangular invertible. Find $\widehat{x}$ and $p$ in terms of $Q$ and $R$ and $b$ (not $A$).

(c) (Separate question) If $q_{1}$ and $q_{2}$ are any orthonormal vectors in $R^5$, give a formula for the projection $p$ of any vector $b$ onto the plane spanned by $q_{1}$ and $q_{2}$ (write $p$ as a combination of $q_{1}$ and $q_{2}$).

(40 points) This problem is about the $n$ by $n$ matrix $A_n$ that has zeros on its main diagonal and all other entries equal to $-1$. In MATLAB $A_n=\text{eye}(n)-\text{ones}(n)$.

(a) Find the determinant of $A_n$. Here is a suggested approach:

Start by adding all rows (except the last) to the last row, and then factoring out a constant. (You could check $n=3$ to have a start on part b.)

(b) For any invertible matrix $A$, the $(1, 1)$ entry of $A^{-1}$ is the ratio of $\underline{\hspace{2cm}}$.

So the $(1, 1)$ entry of $A_4^{-1}$ is $\underline{\hspace{2cm}}$.

(c) Find two orthogonal eigenvectors with $A_3 x=x$. (So $\lambda=1$ is a double eigenvalue.)

(d) What is the third eigenvalue of $A_3$ and a corresponding eigenvector?