MIT 18.06 习题集 2

This lesson reproduces the Problem Set “MIT 18.06 Problem Set 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 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.

Use Gauss-Jordan elimination on $[U \ I]$ to find the upper triangular $U^{-1}$:

(Recommended) $A$ is a 4 by 4 matrix with 1's on the diagonal and $-a,-b,-c$ on the diagonal above. Find $A^{-1}$ for this bidiagonal matrix.

(Recommended) Compute $L$ and $U$ for the symmetric matrix

Find four conditions on $a, b, c, d$ to get $A=LU$ with four pivots.

If $A=LDU$ and also $A=L_1 D_1 U_1$ with all factors invertible, then $L=L_1$ and $D=D_1$ and $U=U_1$. "The three factors are unique."

Derive the equation $L_1^{-1} L D=D_1 U_1 U^{-1}$. Are the two sides triangular or diagonal? Deduce $L=L_1$ and $U=U_1$ (they all have diagonal 1's). Then $D=D_1$.

For the 6 by 6 second difference constant-diagonal matrix $K$, put the pivots and multipliers into $K=LU$. (L and U will have only two nonzero diagonals, because K has three.) Find a formula for the $i, j$ entry of $L^{-1}$, by software like MATLAB using inv(L) or by looking for a nice pattern.

If you print $K^{-1}$, it doesn't look good. But if you print $7K^{-1}$ (when $K$ is 6 by 6), that matrix looks wonderful. Write down $7K^{-1}$ by hand, following this pattern:

1 Row 1 and column 1 are (6, 5, 4, 3, 2, 1).

2 On and above the main diagonal, row $i$ is $i$ times row 1.

3 On and below the main diagonal, column $j$ is $j$ times column 1.

Multiply $K$ times that $7K^{-1}$ to produce $7I$. Here is that pattern for $n=3$ (3 by 3 case; the determinant of this $K$ is 4):

(a) Find a 3 by 3 permutation matrix with $P^3=I$ (but not $P=I$). (b) Find a 4 by 4 permutation $\widehat{P}$ with $\widehat{P}^4 \neq I$.

A group of matrices includes $AB$ and $A^{-1}$ if it includes $A$ and $B$. "Products and inverses stay in the group." Which of these sets are groups?

Lower triangular matrices $L$ with 1's on the diagonal, symmetric matrices $S$, positive matrices $M$, diagonal invertible matrices $D$, permutation matrices $P$, matrices with $Q^{T}=Q^{-1}$. \emph{Invent two more matrix groups.}

Suppose $Q^{T}$ equals $Q^{-1}$ (transpose equal inverse, so $Q^{T}Q=I$).

(a) Show that the columns $q_1, \dots, q_n$ are unit vectors: $\|q_i\|^2=1$.

(b) Show that every two distinct columns of $Q$ are perpendicular: $q_i^{T}q_j=0$ for $i \neq j$.

(c) Find a 2 by 2 example with first entry $q_{11}=\cos\theta$.

True or false (check addition or give a counterexample):

(a) The symmetric matrices in $\mathbf{M}$ (with $A^{T}=A$) form a subspace.

(b) The skew-symmetric matrices in $\mathbf{M}$ (with $A^{T}=-A$) form a subspace.

(c) The unsymmetric matrices in $\mathbf{M}$ (with $A^{T} \neq A$) form a subspace.

(Recommended) If we add an extra column $b$ to a matrix $A$, then the column space gets larger unless $\underline{\hspace{2cm}}$. Give an example where the column space gets larger and an example where it doesn't. Why is $Ax=b$ solvable exactly when the column space \emph{doesn't} get larger—it is the same for $A$ and $[A \ b]$?

Suppose $\mathbf{S}$ and $\mathbf{T}$ are two subspaces of a vector space $\mathbf{V}$.

(a) \textbf{Definition:} The sum $\mathbf{S}+\mathbf{T}$ contains all sums $\mathbf{s}+\mathbf{t}$ of a vector $\mathbf{s}$ in $\mathbf{S}$ and a vector $\mathbf{t}$ in $\mathbf{T}$. Show that $\mathbf{S}+\mathbf{T}$ satisfies the requirements (addition and scalar multiplication) for a vector space.

(b) If $\mathbf{S}$ and $\mathbf{T}$ are lines in $\mathbf{R}^m$, what is the difference between $\mathbf{S}+\mathbf{T}$ and $\mathbf{S} \cup \mathbf{T}$? That union contains all vectors from $\mathbf{S}$ and $\mathbf{T}$ or both. Explain this statement: \emph{The span of} $\mathbf{S} \cup \mathbf{T}$ \emph{is} $\mathbf{S}+\mathbf{T}$\emph{.} (Section 3.5 returns to this word "span.")

Show that the matrices $A$ and $[A \ AB]$ (with extra columns) have the same column space. But find a square matrix with $\mathbf{C}(A^2)$ smaller than $\mathbf{C}(A)$. \textbf{Important point:}

An $n$ by $n$ matrix has $\mathbf{C}(A)=\mathbf{R}^n$ exactly when $A$ is an $\underline{\hspace{2cm}}$ matrix.