MIT 线性代数习题

MIT 习题课地址

0、线性代数中的几何学

Solve
$\begin{aligned} 2x + y &= 3 \\ x -2y &=-1 \end{aligned}$2x+yx−2y​=3=−1​
and find out its “row picture” and “column picture”

1、核心思想概述

Suppose $A$A is a matrix such that the complete solution to $Ax=\begin{bmatrix}1\\4\\1\\1\end{bmatrix}$Ax=⎣⎢⎢⎡​1411​⎦⎥⎥⎤​ is $x=\begin{bmatrix}0\\1\\1\end{bmatrix}+c\begin{bmatrix}0\\2\\1\end{bmatrix}$x=⎣⎡​011​⎦⎤​+c⎣⎡​021​⎦⎤​，what can you say about columns of $A$A？

2、矩阵的消去法

Solve using the method of elimination:
$\begin{aligned} x-y-z+u&=0\\ 2x+2z&=8\\ -y-2z&=-8\\ 3x-3y-2z+4u&=7 \end{aligned}$x−y−z+u2x+2z−y−2z3x−3y−2z+4u​=0=8=−8=7​
3、逆矩阵

Find the conditions on $a$a and $b$b that make the matrix A invertible, and find $A^{-1}$A−1 when it exists.
$A=\begin{bmatrix}a&b&b\\a&a&b\\a&a&a\end{bmatrix}$A=⎣⎡​aaa​baa​bba​⎦⎤​
4、LU分解

Find the LU-decomposition of the matrix $A$A when it exists. For which real numbers $a$a and $b$b does it exist?
$A=\begin{bmatrix}1&0&1\\a&a&a\\b&b&a\end{bmatrix}$A=⎣⎡​1ab​0ab​1aa​⎦⎤​

5、三维空间的子空间

$x_1=\begin{bmatrix}0\\1\\3\end{bmatrix}$x1​=⎣⎡​013​⎦⎤​, $x_2=\begin{bmatrix}2\\4\\0\end{bmatrix}$x2​=⎣⎡​240​⎦⎤​

• Find subspace $V_1$V1​ generated by $x_1$x1​，subspace $V_2$V2​ generated by $x_2$x2​，Describe $V_1 \cap V_2$V1​∩V2​
• Find subspace $V_3$V3​ generated by $\begin{bmatrix}x_1&x_2 \end{bmatrix}$[x1​​x2​​], Is $V_3$V3​ equal to $V_1\cup V_2$V1​∪V2​? Find a subspace $S$S of $V_3$V3​ such that $x_1 \notin S, x_2 \notin S$x1​∈/​S,x2​∈/​S.
• What is $V_3 \cap \{xy\mathbb{ plane}\}$V3​∩{xyplane} ?

6、向量子空间

Which are subspaces of $\mathbb{R}^3 = \{\begin{bmatrix} l_1\\l_2\\l_3\end{bmatrix}\}$R3={⎣⎡​l1​l2​l3​​⎦⎤​}

1. $l_1+l_2-l_3=0$l1​+l2​−l3​=0

2. $l_1l_2-l_3=0$l1​l2​−l3​=0

3. $\begin{bmatrix} l_1\\l_2\\l_3\end{bmatrix}=\begin{bmatrix} 1\\0\\0\end{bmatrix}+c_1\begin{bmatrix} 1\\0\\-1\end{bmatrix}+c_2\begin{bmatrix} 1\\0\\1\end{bmatrix}$⎣⎡​l1​l2​l3​​⎦⎤​=⎣⎡​100​⎦⎤​+c1​⎣⎡​10−1​⎦⎤​+c2​⎣⎡​101​⎦⎤​

4. $\begin{bmatrix} l_1\\l_2\\l_3\end{bmatrix}=\begin{bmatrix} 0\\1\\0\end{bmatrix}+c_1\begin{bmatrix} 1\\0\\-1\end{bmatrix}+c_2\begin{bmatrix} 1\\0\\1\end{bmatrix}$⎣⎡​l1​l2​l3​​⎦⎤​=⎣⎡​010​⎦⎤​+c1​⎣⎡​10−1​⎦⎤​+c2​⎣⎡​101​⎦⎤​

7、解$Ax=0$Ax=0

The set $S$S of points $P(x,y,z)$P(x,y,z) such that $x-5y+2z=9$x−5y+2z=9 is a ____ in $\mathbb{R}^3$R3. It is ____ to the ____ $S_0$S0​ of $P(x,y,z)$P(x,y,z) such that $x-5y+2z=0$x−5y+2z=0

8、解 $Ax=b$Ax=b

Find all solutions, depending on $b_1$b1​，$b_2$b2​，$b_3$b3​：
$\begin{aligned} x-2y-2z&=b_1\\ 2x-5y-4z&=b_2 \\ 4x-9y-8z&=b_3 \end{aligned}$x−2y−2z2x−5y−4z4x−9y−8z​=b1​=b2​=b3​​
9、向量空间的基底与维数

Find the dimension of the vector space spanned by the following vectors
$\begin{bmatrix}1&1&-2&0&-1\end{bmatrix}\\ \begin{bmatrix}1&2&0&-4&1\end{bmatrix}\\ \begin{bmatrix}0&1&3&-3&2\end{bmatrix}\\ \begin{bmatrix}2&3&0&-2&0\end{bmatrix}$[1​1​−2​0​−1​][1​2​0​−4​1​][0​1​3​−3​2​][2​3​0​−2​0​]
and find a basis for that space.

10、四个基本子空间的计算

Suppose $B=\begin{bmatrix}1&&\\2&1\\-1&0&1\end{bmatrix}$B=⎣⎡​12−1​10​1​⎦⎤​$\begin{bmatrix}5&0&3\\0&1&1\\0&0&0\end{bmatrix}$⎣⎡​500​010​310​⎦⎤​

Find a basis for and compute the dimension of each of the 4 fundamental subspaces of $B$B.

11、矩阵的空间

Show that the set of $2 \times 3$2×3 matrices whose null space contains $\begin{bmatrix}2\\1\\1\end{bmatrix}$⎣⎡​211​⎦⎤​ is a vector subspace, and find a basis for it. What about the set of those whose column space contains $\begin{bmatrix}2\\1\end{bmatrix}$[21​].

12、测验题目讲解1

$A=\begin{bmatrix}1&1&1\\1&2&3\\3&4&k\end{bmatrix}$A=⎣⎡​113​124​13k​⎦⎤​

a) For which $k$k does $Ax=\begin{bmatrix}2\\3\\7\end{bmatrix}$Ax=⎣⎡​237​⎦⎤​ have a unique solution

b) Which $k$k, does $Ax$Ax has infinitely many solution.

c) When $k=4$k=4, find LU decomposition.

d) For all $k$k, find complete solution.

13、图像与网络

• Find incidence matrix $A$A

• $N(A)$N(A), $N(A^T)$N(AT)

• $\mathbb{Tr}(A^TA)$Tr(ATA)

14、正交向量和子空间

$S$S is spanned by $\begin{bmatrix}1&2&2&3\end{bmatrix}$[1​2​2​3​] and $\begin{bmatrix}1&3&3&2\end{bmatrix}$[1​3​3​2​].

1. Find a basis for $S^{\perp}$S⊥

2. Can every $v$v in $\mathbb{R}^4$R4 be written uniquely in terms of $S$S and $S^{\perp} $

15、子空间上的投影

Find the orthogonal projection matrix onto the plane: $x+y-z=0$x+y−z=0

16、最小二乘逼近

Find the quadratic equation through the origin that is a best fit for the points $(1,1), (2,5),(-1,-2)$(1,1),(2,5),(−1,−2)

17、Gram-Schmidt 正交化

Find $q_1,q_2,q_3$q1​,q2​,q3​( orthogonal) from columns of $A$A. Then write $A$A as $QR$QR ($Q$Q orthogonal, $R$R upper triangular)

$A=\begin{bmatrix}1&2&4\\0&0&5\\0&3&6\end{bmatrix}$A=⎣⎡​100​203​456​⎦⎤​

18、行列式的性质

Find the determinants of

$A=\begin{bmatrix}101&201&301\\102&202&302\\103&203&303\end{bmatrix}$A=⎣⎡​101102103​201202203​301302303​⎦⎤​

$B=\begin{bmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{bmatrix}$B=⎣⎡​111​abc​a2b2c2​⎦⎤​

$C=\begin{bmatrix}1\\2\\3 \end{bmatrix}\begin{bmatrix}1&-4&5 \end{bmatrix}$C=⎣⎡​123​⎦⎤​[1​−4​5​]

$D=\begin{bmatrix}0&1&3\\-1&0&4\\-3&-4&0\end{bmatrix}$D=⎣⎡​0−1−3​10−4​340​⎦⎤​

19、行列式

Find the determinants of

$A=\begin{bmatrix}x&y&0&0&0\\0&x&y&0&0\\0&0&x&y&0\\0&0&0&x&y\\y&0&0&0&x\end{bmatrix}$A=⎣⎢⎢⎢⎢⎡​x000y​yx000​0yx00​00yx0​000yx​⎦⎥⎥⎥⎥⎤​

$B=\begin{bmatrix}x&y&y&y&y\\y&x&y&y&y\\y&y&x&y&y\\y&y&y&x&y\\y&y&y&y&x\end{bmatrix}$B=⎣⎢⎢⎢⎢⎡​xyyyy​yxyyy​yyxyy​yyyxy​yyyyx​⎦⎥⎥⎥⎥⎤​

Hint: You may combine two of the methods: (1) Elimination (2) $\sum\pm a_{1\alpha}a_{2\beta}a_{3\gamma}$∑±a1α​a2β​a3γ​ (3) By cofactors

20、行列式与体积

$T$T is a tetrahedron with vertex $O(0,0,0), A_1(2,2,-1),A_2(1,3,0),A_3(-1,1,4)$O(0,0,0),A1​(2,2,−1),A2​(1,3,0),A3​(−1,1,4), Compute Vol(T). If $A_1$A1​, $A_2$A2​ are fixed, but $A_3$A3​ is moved to $A_3^{'}(-201,-199,104)$A3′​(−201,−199,104), compute Vol(T) again.

21、特征值和特征向量

Given the invertible $A=\begin{bmatrix}1&2&3\\0&1&-2\\0&1&4\end{bmatrix}$A=⎣⎡​100​211​3−24​⎦⎤​, find the eigenvalues and eigenvectors of $A^2$A2, $A^{-1}-I$A−1−I

22、矩阵的方幂

Find a formula for $C^k$Ck where $C=\begin{bmatrix}2b-a&a-b \\2b-2a&2a-b\end{bmatrix}$C=[2b−a2b−2a​a−b2a−b​], calculate $C^{100}$C100 when $a=b=-1$a=b=−1

23、微分方程与exp(At)

Solve the differential equation $y^{'''}+2y^{''}-y^{'}-2y=0$y′′′+2y′′−y′−2y=0 for the general solution. What is the matrix $A$A. Find the first column of exp($At$At)

24、马尔科夫矩阵

A particle jumps between positions A and B with the following probabilities.

If it starts at A, what is the probability it is at A and B after i) 1 step ii) n steps iii) $\infin$∞ steps

25、测验题目讲解2

$A=\begin{bmatrix}1&2&3&4\\5&6&7&8\\0&0&9&10\\0&0&11&12\end{bmatrix}$A=⎣⎢⎢⎡​1500​2600​37911​481012​⎦⎥⎥⎤​

1. Find all the non-zero terms in the big formula $\mathbb{det}A=\sum\pm a_{1\alpha}a_{2\beta}a_{3\gamma}a_{4\delta}$detA=∑±a1α​a2β​a3γ​a4δ​ and compute $\mathbb{det}A $
2. Find cofactors $C_{11}$C11​,$C_{12}$C12​,$C_{13}$C13​ and $C_{14}$C14​
3. Find column 1 of $A^{-1}$A−1

26、对称矩阵与正定矩阵

Explain why each of the following is true.

a) Every positive definite matrix is invertible

b) The only positive definite projection matrix is $P = I$P=I

c) D is diagonal with positive entries is positive definite

d) $S$S symmetric with $\mathbb{det}S>0$detS>0 might not be positive definite

27、复矩阵

Diagonalize $A$A by constructing its eigenvalue matrix $\Lambda$Λ and eigenvector matrix $S$S

$A=\begin{bmatrix}2&1-i\\1+i&3\end{bmatrix}=\bar{A}^T=A^H$A=[21+i​1−i3​]=AˉT=AH

28、正定矩阵与极小值

For which values of c is
$B=\begin{bmatrix}2&-1&-1\\-1&2&-1\\-1&-1&2+c\end{bmatrix}$B=⎣⎡​2−1−1​−12−1​−1−12+c​⎦⎤​

• positive definite?
• positive semidefinite?

29、相似矩阵

Which of the following statements are true? Explain

(a) If $A$A and $B$B are similar matrices, then $2A^3+A-3I$2A3+A−3I and $2B^3+B-3I$2B3+B−3I are similar

(b) If $A$A and $B$B are $3 \times 3$3×3 matrices with eigenvalues 1,0,-1， then $A$A and $B$B are similar.

© The matrices $J_1=\begin{bmatrix}-1&1&0\\0&-1&1\\0&0&-1\end{bmatrix}$J1​=⎣⎡​−100​1−10​01−1​⎦⎤​ and $J_2=\begin{bmatrix}-1&1&0\\0&-1&0\\0&0&-1\end{bmatrix}$J2​=⎣⎡​−100​1−10​00−1​⎦⎤​ are similar

30、奇异值分解的运算

Find the singular value decomposition of the matrix $C=\begin{bmatrix}5&5\\-1&7\end{bmatrix}$C=[5−1​57​]

31、线性变换

Let $T(A)=A^T$T(A)=AT, $A$A is $2 \times 2$2×2

1. why is T linear? What is T^{-1}?

2. Write down the matrix of T in

$v_1=\begin{bmatrix}1&0\\0&0\end{bmatrix}$v1​=[10​00​], $v_2=\begin{bmatrix}0&1\\0&0\end{bmatrix}$v2​=[00​10​], $v_3=\begin{bmatrix}0&0\\1&0\end{bmatrix}$v3​=[01​00​],$v_4=\begin{bmatrix}0&0\\0&1\end{bmatrix}$v4​=[00​01​]

$w_1=\begin{bmatrix}1&0\\1&0\end{bmatrix}$w1​=[11​00​], $v_2=\begin{bmatrix}0&0\\0&1\end{bmatrix}$v2​=[00​01​], $v_3=\begin{bmatrix}0&1\\1&0\end{bmatrix}$v3​=[01​10​],$v_4=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$v4​=[0−1​10​]

3） Eigenvalues / eigenvectors of T?

32、基的变换

the vector space of all polynomials in x of degree $\leq 2$≤2 has a basis $1,x,x^2$1,x,x2. Let $\omega_1,\omega_2,\omega_3$ω1​,ω2​,ω3​ be a different basis of polynomials whose values at x = -1,-,1 are given by:

a) Express $y(x)=-x+5$y(x)=−x+5 in this basis

b) Find the change of basis matrices ($1$1,$x$x,$x^2$x2) $\leftrightarrow (w_1,w_2,w_3)$↔(w1​,w2​,w3​)

c) Find the matrix of taking derivatives in both basis

33、广义逆

Given $A=\begin{bmatrix}1&2\end{bmatrix}$A=[1​2​]

i) What is $A^+$A+(pseudoinverse)

ii) $AA^+$AA+ and $A^+A$A+A

iii) If $x$x is in $N(A)$N(A), what is $A^+Ax$A+Ax

iv) If $x$x is in $C(A^T)$C(AT), what is $A^+Ax$A+Ax

34、测验题目讲解3

Find the eigenvalues and eigenvectors of the following

i) Projection $P=\frac{aa^T}{a^Ta}$P=aTaaaT​, $a=\begin{bmatrix}3\\4\end{bmatrix}$a=[34​]

ii) $Q=\begin{bmatrix}0.6&-0.8\\0.8&0.6\end{bmatrix}$Q=[0.60.8​−0.80.6​]

iii) $R=2P-I$R=2P−I

35、期末考试题讲解

$A=\begin{bmatrix}1&0&1\\0&1&1\\1&1&0\end{bmatrix}$A=⎣⎡​101​011​110​⎦⎤​. Two eigenvalues: $\lambda_1=1$λ1​=1, $\lambda_2=2$λ2​=2, First two pivots: $d_1=d_2=1$d1​=d2​=1

(a) Find $\lambda_3$λ3​ and $d_3$d3​

(b) What is the smallest $a_{33}$a33​ that would make $A$A positive semidefinite? What is the smallest c that $A+cI$A+cI is positive semi-definite？