MIT 18.06 期末考试

This lesson reproduces the Exam “MIT 18.06 Final Exam” 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 9 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.

This question is about the matrix

(a) Find a lower triangular $L$ and an upper triangular $U$ so that $A=LU$.

(b) Find the reduced row echelon form $R=rref(A)$. How many independent columns in $A$?

(c) Find a basis for the nullspace of $A$.

(d) If the vector $b$ is the sum of the four columns of $A$, write down the complete solution to $Ax=b$.

This problem finds the curve $y=C+D2^t$ which gives the best least squares fit to the points $(t, y)=(0, 6), (1, 4), (2, 0)$.

(a) Write down the 3 equations that would be satisfied if the curve went through all 3 points.

(b) Find the coefficients $C$ and $D$ of the best curve $y=C+D2^t$.

(c) What values should $y$ have at times $t=0, 1, 2$ so that the best curve is $y=0$?

Suppose $Av_i=b_i$ for the vectors $v_1, \ldots, v_n$ and $b_1, \ldots, b_n$ in $\mathbb{R}^n$. Put the $v$'s into the columns of $V$ and put the $b$'s into the columns of $B$.

(a) Write those equations $Av_i=b_i$ in matrix form. What condition on which vectors allows $A$ to be determined uniquely? Assuming this condition, find $A$ from $V$ and $B$.

(b) Describe the column space of that matrix $A$ in terms of the given vectors.

(c) What additional condition on which vectors makes $A$ an invertible matrix? Assuming this, find $A^{-1}$ from $V$ and $B$.

(a) Suppose $x_k$ is the fraction of MIT students who prefer calculus to linear algebra at year $k$. The remaining fraction $y_k=1-x_k$ prefers linear algebra.

At year $k+1$, $1/5$ of those who prefer calculus change their mind (possibly after taking 18.03). Also at year $k+1$, $1/10$ of those who prefer linear algebra change their mind (possibly because of this exam).

Create the matrix $A$ to give $\begin{bmatrix} x_{k+1} \\ y_{k+1} \end{bmatrix}=A \begin{bmatrix} x_k \\ y_k \end{bmatrix}$ and find the limit of $A^k \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ as $k \to \infty$.

(b) Solve these differential equations, starting from $x(0)=1$, $y(0)=0$:

(c) For what initial conditions $\begin{bmatrix} x(0) \\ y(0) \end{bmatrix}$ does the solution $\begin{bmatrix} x(t) \\ y(t) \end{bmatrix}$ to this differential equation lie on a single straight line in $\mathbb{R}^2$ for all $t$?

(a) Consider a $120^\circ$ rotation around the axis $x=y=z$. Show that the vector $i=(1, 0, 0)$ is rotated to the vector $j=(0, 1, 0)$. (Similarly $j$ is rotated to $k=(0, 0, 1)$ and $k$ is rotated to $i$.) How is $j-i$ related to the vector $(1, 1, 1)$ along the axis?

(b) Find the matrix $A$ that produces this rotation (so $Av$ is the rotation of $v$). Explain why $A^3=I$. What are the eigenvalues of $A$?

(c) If a 3 by 3 matrix $P$ projects every vector onto the plane $x+2y+z=0$, find three eigenvalues and three independent eigenvectors of $P$. No need to compute $P$.

This problem is about the matrix

(a) Find the eigenvalues of $A^T A$ and also of $AA^T$. For both matrices find a complete set of orthonormal eigenvectors.

(b) If you apply the Gram-Schmidt process (orthonormalization) to the columns of this matrix $A$, what is the resulting output?

(c) If $A$ is any $m$ by $n$ matrix with $m > n$, tell me why $AA^T$ cannot be positive definite. Is $A^T A$ always positive definite? (If not, what is the test on $A$?)

This problem is to find the determinants of

(a) Find det $A$ and give a reason.

(b) Find the cofactor $C_{11}$ and then find det $B$. This is the volume of what region in $\mathbb{R}^4$?

(c) Find det $C$ for any value of $x$. You could use linearity in row 1.

(a) When $A$ is similar to $B=M^{-1}AM$, prove this statement:

If $A^k \to 0$ when $k \to \infty$, then also $B^k \to 0$.

(b) Suppose $S$ is a fixed invertible 3 by 3 matrix.

This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of 3 by 3 matrix space. (Test the requirements for a subspace.)

(c) Give a basis for the space of 3 by 3 diagonal matrices. Find a basis for the space in part (b) — all the matrices $A$ that are diagonalized by $S$.

This square network has 4 nodes and 6 edges. On each edge, the direction of positive current $w_i > 0$ is from lower node number to higher node number. The voltages at the nodes are $(v_1, v_2, v_3, v_4)$.


(a) Write down the incidence matrix $A$ for this network (so that $Av$ gives the 6 voltage differences like $v_2-v_1$ across the 6 edges). What is the rank of $A$? What is the dimension of the nullspace of $A^T$?

(b) Compute the matrix $A^T A$. What is its rank? What is its nullspace?

(c) Suppose $v_1=1$ and $v_4=0$. If each edge contains a unit resistor, the currents $(w_1, w_2, w_3, w_4, w_5, w_6)$ on the 6 edges will be $w=-Av$ by Ohm's Law. Then Kirchhoff's Current Law (flow in=flow out at every node) gives $A^T w=0$ which means $A^T A v=0$. Solve $A^T A v=0$ for the unknown voltages $v_2$ and $v_3$. Find all 6 currents $w_1$ to $w_6$. How much current enters node 4?