MIT 线性代数习题

MIT 习题课地址

0、线性代数中的几何学

Solve
2x+y=3x−2y=−1\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 AAA is a matrix such that the complete solution to Ax=[1411]Ax=\begin{bmatrix}1\\4\\1\\1\end{bmatrix}Ax=⎣⎢⎢⎡1411⎦⎥⎥⎤ is x=[011]+c[021]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 AAA？

2、矩阵的消去法

Solve using the method of elimination:
x−y−z+u=02x+2z=8−y−2z=−83x−3y−2z+4u=7\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 aaa and bbb that make the matrix A invertible, and find A−1A^{-1}A−1 when it exists.
A=[abbaabaaa]A=\begin{bmatrix}a&b&b\\a&a&b\\a&a&a\end{bmatrix} A=⎣⎡aaabaabba⎦⎤
4、LU分解

Find the LU-decomposition of the matrix AAA when it exists. For which real numbers aaa and bbb does it exist?
A=[101aaabba]A=\begin{bmatrix}1&0&1\\a&a&a\\b&b&a\end{bmatrix} A=⎣⎡1ab0ab1aa⎦⎤

5、三维空间的子空间

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

Find subspace V1V_1V1 generated by x1x_1x1，subspace V2V_2V2 generated by x2x_2x2，Describe V1∩V2V_1 \cap V_2V1∩V2
Find subspace V3V_3V3 generated by [x1x2]\begin{bmatrix}x_1&x_2 \end{bmatrix}[x1x2], Is V3V_3V3 equal to V1∪V2V_1\cup V_2V1∪V2? Find a subspace SSS of V3V_3V3 such that x1∉S,x2∉Sx_1 \notin S, x_2 \notin Sx1∈/S,x2∈/S.
What is V3∩{xyplane}V_3 \cap \{xy\mathbb{ plane}\}V3∩{xyplane} ?

6、向量子空间

Which are subspaces of R3={[l1l2l3]}\mathbb{R}^3 = \{\begin{bmatrix} l_1\\l_2\\l_3\end{bmatrix}\}R3={⎣⎡l1l2l3⎦⎤}

l1+l2−l3=0l_1+l_2-l_3=0l1+l2−l3=0
l1l2−l3=0l_1l_2-l_3=0l1l2−l3=0
[l1l2l3]=[100]+c1[10−1]+c2[101]\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}⎣⎡l1l2l3⎦⎤=⎣⎡100⎦⎤+c1⎣⎡10−1⎦⎤+c2⎣⎡101⎦⎤
[l1l2l3]=[010]+c1[10−1]+c2[101]\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}⎣⎡l1l2l3⎦⎤=⎣⎡010⎦⎤+c1⎣⎡10−1⎦⎤+c2⎣⎡101⎦⎤

7、解Ax=0Ax=0Ax=0

The set SSS of points P(x,y,z)P(x,y,z)P(x,y,z) such that x−5y+2z=9x-5y+2z=9x−5y+2z=9 is a ____ in R3\mathbb{R}^3R3. It is ____ to the ____ S0S_0S0 of P(x,y,z)P(x,y,z)P(x,y,z) such that x−5y+2z=0x-5y+2z=0x−5y+2z=0

8、解 Ax=bAx=bAx=b

Find all solutions, depending on b1b_1b1，b2b_2b2，b3b_3b3：
x−2y−2z=b12x−5y−4z=b24x−9y−8z=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
[11−20−1][120−41][013−32][230−20]\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} [11−20−1][120−41][013−32][230−20]
and find a basis for that space.

10、四个基本子空间的计算

Suppose B=[121−101]B=\begin{bmatrix}1&&\\2&1\\-1&0&1\end{bmatrix}B=⎣⎡12−1101⎦⎤[503011000]\begin{bmatrix}5&0&3\\0&1&1\\0&0&0\end{bmatrix}⎣⎡500010310⎦⎤

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

11、矩阵的空间

Show that the set of 2×32 \times 32×3 matrices whose null space contains [211]\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 [21]\begin{bmatrix}2\\1\end{bmatrix}[21].

12、测验题目讲解1

A=[11112334k]A=\begin{bmatrix}1&1&1\\1&2&3\\3&4&k\end{bmatrix}A=⎣⎡11312413k⎦⎤

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

b) Which kkk, does AxAxAx has infinitely many solution.

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

d) For all kkk, find complete solution.

13、图像与网络

Find incidence matrix AAA
N(A)N(A)N(A), N(AT)N(A^T)N(AT)
Tr(ATA)\mathbb{Tr}(A^TA)Tr(ATA)

14、正交向量和子空间

SSS is spanned by [1223]\begin{bmatrix}1&2&2&3\end{bmatrix}[1223] and [1332]\begin{bmatrix}1&3&3&2\end{bmatrix}[1332].

Find a basis for S⊥S^{\perp}S⊥
Can every vvv in R4\mathbb{R}^4R4 be written uniquely in terms of SSS and $S^{\perp} $

15、子空间上的投影

Find the orthogonal projection matrix onto the plane: x+y−z=0x+y-z=0x+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)(1,1),(2,5),(−1,−2)

17、Gram-Schmidt 正交化

Find q1,q2,q3q_1,q_2,q_3q1,q2,q3( orthogonal) from columns of AAA. Then write AAA as QRQRQR (QQQ orthogonal, RRR upper triangular)

A=[124005036]A=\begin{bmatrix}1&2&4\\0&0&5\\0&3&6\end{bmatrix}A=⎣⎡100203456⎦⎤

18、行列式的性质

Find the determinants of

A=[101201301102202302103203303]A=\begin{bmatrix}101&201&301\\102&202&302\\103&203&303\end{bmatrix}A=⎣⎡101102103201202203301302303⎦⎤

B=[1aa21bb21cc2]B=\begin{bmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{bmatrix}B=⎣⎡111abca2b2c2⎦⎤

C=[123][1−45]C=\begin{bmatrix}1\\2\\3 \end{bmatrix}\begin{bmatrix}1&-4&5 \end{bmatrix}C=⎣⎡123⎦⎤[1−45]

D=[013−104−3−40]D=\begin{bmatrix}0&1&3\\-1&0&4\\-3&-4&0\end{bmatrix}D=⎣⎡0−1−310−4340⎦⎤

19、行列式

Find the determinants of

A=[xy0000xy0000xy0000xyy000x]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=⎣⎢⎢⎢⎢⎡x000yyx0000yx0000yx0000yx⎦⎥⎥⎥⎥⎤

B=[xyyyyyxyyyyyxyyyyyxyyyyyx]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=⎣⎢⎢⎢⎢⎡xyyyyyxyyyyyxyyyyyxyyyyyx⎦⎥⎥⎥⎥⎤

Hint: You may combine two of the methods: (1) Elimination (2) ∑±a1αa2βa3γ\sum\pm a_{1\alpha}a_{2\beta}a_{3\gamma}∑±a1αa2βa3γ (3) By cofactors

20、行列式与体积

TTT is a tetrahedron with vertex O(0,0,0),A1(2,2,−1),A2(1,3,0),A3(−1,1,4)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 A1A_1A1, A2A_2A2 are fixed, but A3A_3A3 is moved to A3′(−201,−199,104)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=[12301−2014]A=\begin{bmatrix}1&2&3\\0&1&-2\\0&1&4\end{bmatrix}A=⎣⎡1002113−24⎦⎤, find the eigenvalues and eigenvectors of A2A^2A2, A−1−IA^{-1}-IA−1−I

22、矩阵的方幂

Find a formula for CkC^kCk where C=[2b−aa−b2b−2a2a−b]C=\begin{bmatrix}2b-a&a-b \\2b-2a&2a-b\end{bmatrix}C=[2b−a2b−2aa−b2a−b], calculate C100C^{100}C100 when a=b=−1a=b=-1a=b=−1

23、微分方程与exp(At)

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

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=[1234567800910001112]A=\begin{bmatrix}1&2&3&4\\5&6&7&8\\0&0&9&10\\0&0&11&12\end{bmatrix}A=⎣⎢⎢⎡1500260037911481012⎦⎥⎥⎤

Find all the non-zero terms in the big formula detA=∑±a1αa2βa3γa4δ\mathbb{det}A=\sum\pm a_{1\alpha}a_{2\beta}a_{3\gamma}a_{4\delta}detA=∑±a1αa2βa3γa4δ and compute $\mathbb{det}A $
Find cofactors C11C_{11}C11,C12C_{12}C12,C13C_{13}C13 and C14C_{14}C14
Find column 1 of A−1A^{-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=IP = IP=I

c) D is diagonal with positive entries is positive definite

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

27、复矩阵

Diagonalize AAA by constructing its eigenvalue matrix Λ\LambdaΛ and eigenvector matrix SSS

A=[21−i1+i3]=AˉT=AHA=\begin{bmatrix}2&1-i\\1+i&3\end{bmatrix}=\bar{A}^T=A^HA=[21+i1−i3]=AˉT=AH

28、正定矩阵与极小值

For which values of c is
B=[2−1−1−12−1−1−12+c]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 AAA and BBB are similar matrices, then 2A3+A−3I2A^3+A-3I2A3+A−3I and 2B3+B−3I2B^3+B-3I2B3+B−3I are similar

(b) If AAA and BBB are 3×33 \times 33×3 matrices with eigenvalues 1,0,-1， then AAA and BBB are similar.

© The matrices J1=[−1100−1100−1]J_1=\begin{bmatrix}-1&1&0\\0&-1&1\\0&0&-1\end{bmatrix}J1=⎣⎡−1001−1001−1⎦⎤ and J2=[−1100−1000−1]J_2=\begin{bmatrix}-1&1&0\\0&-1&0\\0&0&-1\end{bmatrix}J2=⎣⎡−1001−1000−1⎦⎤ are similar

30、奇异值分解的运算

Find the singular value decomposition of the matrix C=[55−17]C=\begin{bmatrix}5&5\\-1&7\end{bmatrix}C=[5−157]

31、线性变换

Let T(A)=ATT(A)=A^TT(A)=AT, AAA is 2×22 \times 22×2

why is T linear? What is T^{-1}?
Write down the matrix of T in

v1=[1000]v_1=\begin{bmatrix}1&0\\0&0\end{bmatrix}v1=[1000], v2=[0100]v_2=\begin{bmatrix}0&1\\0&0\end{bmatrix}v2=[0010], v3=[0010]v_3=\begin{bmatrix}0&0\\1&0\end{bmatrix}v3=[0100],v4=[0001]v_4=\begin{bmatrix}0&0\\0&1\end{bmatrix}v4=[0001]

w1=[1010]w_1=\begin{bmatrix}1&0\\1&0\end{bmatrix}w1=[1100], v2=[0001]v_2=\begin{bmatrix}0&0\\0&1\end{bmatrix}v2=[0001], v3=[0110]v_3=\begin{bmatrix}0&1\\1&0\end{bmatrix}v3=[0110],v4=[01−10]v_4=\begin{bmatrix}0&1\\-1&0\end{bmatrix}v4=[0−110]

3） Eigenvalues / eigenvectors of T?

32、基的变换

the vector space of all polynomials in x of degree ≤2\leq 2≤2 has a basis 1,x,x21,x,x^21,x,x2. Let ω1,ω2,ω3\omega_1,\omega_2,\omega_3ω1,ω2,ω3 be a different basis of polynomials whose values at x = -1,-,1 are given by:

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

b) Find the change of basis matrices (111,xxx,x2x^2x2) ↔(w1,w2,w3)\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=[12]A=\begin{bmatrix}1&2\end{bmatrix}A=[12]

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

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

iii) If xxx is in N(A)N(A)N(A), what is A+AxA^+AxA+Ax

iv) If xxx is in C(AT)C(A^T)C(AT), what is A+AxA^+AxA+Ax

34、测验题目讲解3

Find the eigenvalues and eigenvectors of the following

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

ii) Q=[0.6−0.80.80.6]Q=\begin{bmatrix}0.6&-0.8\\0.8&0.6\end{bmatrix}Q=[0.60.8−0.80.6]

iii) R=2P−IR=2P-IR=2P−I

35、期末考试题讲解

A=[101011110]A=\begin{bmatrix}1&0&1\\0&1&1\\1&1&0\end{bmatrix}A=⎣⎡101011110⎦⎤. Two eigenvalues: λ1=1\lambda_1=1λ1=1, λ2=2\lambda_2=2λ2=2, First two pivots: d1=d2=1d_1=d_2=1d1=d2=1

(a) Find λ3\lambda_3λ3 and d3d_3d3

(b) What is the smallest a33a_{33}a33 that would make AAA positive semidefinite? What is the smallest c that A+cIA+cIA+cI is positive semi-definite？

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