Linear Algebra I & Mathematics Tutorial 1b (menti.com : 6523 2157) Tutorial 2, October 12th 2021, 14:45 - 15:29 Today:

Homework 1 •Recall Lecture 2 •Start looking at Homework 2 •

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Linear Algebra I & Mathematics Tutorial 1bNagoya University, G30 Program

Fall 2021Instructor: Henrik Bachmann

Homework 1: Linear systems•

Deadline: 17th October, 2021

Exercise 1. (2 Points) Try to solve the exercises below and write the solutions down by hand (paper,tablet) or by computer (Latex only. No word!). Write your name, the homework number and the coursename on the first page of your solution. Create one pdf-file (for example, by using a scanner app onyour phone) and submit it before the deadline ends in NUCT at the Assignment ”Homework 1”. Useexactly the following format as a filename: ”Familyname Givenname LA1 HW1.pdf”.

A linear system is said to be on row-reduced echelon form if the following three conditions aresatisfied:

(i) The first (that is, the leftmost) variable in each equation has coefficient 1.

(ii) If xi is the first variable in one of the equations, then it does not occur in any other equation in thesystem.

(iii) If xi is the first variable in one equation, then the equations below it do not contain any of thevariables x1, x2, . . . , xi−1.

Exercise 2. (2+2+2+2 = 8 Points) Which of the following linear systems are on row-reduced echelonform? For those that are not, find an equivalent system (i.e. one which has the same solutions) that ison row-reduced echelon form. For each system, find all solutions.

i)

x1 − 2x2 + x3 = 73x1 + 2x3 = 1

−2x1 + x2 + 4x3 = 7

ii) x1 + 2x2 + 3x3 + 4x4 = 5

iii)

{x1 + x2 + x3 + 2x4 = 0

x2 + x4 = 0

iv)

{x1 + 2x2 = 3

4x1 + 8x2 = 16

Exercise 3. (8 Points) A Japanese restaurant in 　やごと八事　 (Yagoto, a neighbourhood in Nagoya) is holdingan Ebi Festival, and thus is only selling three types of dishes: Ebi Sushi (Y=370), Ebi Tempura Don(Y=590), and Ebi Fry Bento (Y=830). One serving of Ebi Sushi requires 3 ounces of shrimp, 1 cup of rice,

and 3 tablespoon of shouyu. 5 ounces of shrimp, 4 cups of rice, and 52 tablespoons of shouyu are needed

for one portion of Ebi Tempura Don. For one serving of Ebi Fry Bento, 8 ounces of shrimp, 3 cups ofrice, and 1

2 tablespoons of shouyu are needed. In one certain day, the store expended 1000 ounces ofshrimp, 500 cups of rice, and 500 tablespoons of shouyu.

The market prices are: Y=50 per ounce of shrimp, Y=30 per cup of rice, and Y=5 per tablespoon of shouyu.Given all these information, how much profit did the restaurant make on this certain day? Describe thisproblem by using a linear system, bring the linear system on row-reduced echelon form and solve it.

Exercise 4. (6 Points) Decide for which real numbers a ∈ R the following linear system has solutions.Give all the solutions in these cases.

(a− 1)2x1 + x2 + ax3 = 0x1 + x2 = 02x1 + 3x2 + x3 = a

.

Version: October 3, 2021- 1 -

Recall Lecture 2

Linear Algebra I • Matrices & Vectors

Definition 2.2. For matrices A = (aij), B = (bij) ∈ Rm×n and a real number λ ∈ R we define

A+B = (aij + bij) ∈ Rm×n (Sum of two matrices) ,

λA = (λaij) ∈ Rm×n (Scalar multiplication) .

In the case λ = −1 we write (−1)A = −A and A−B means A+ (−1)B.

The matrices A and B need to be of the same size, otherwise the sum A+B is not defined. A specialcase of the addition of matrices is given by the addition of vectors. For u, v ∈ Rn and λ ∈ R we have

u =

u1...un

, v =

v1...vn

, u+ v =

u1 + v1

...un + vn

, λv =

λv1...

λvn

.

Definition 2.3. The product of a matrix A = (aij) ∈ Rm×n and a vector v ∈ Rn is defined by

Av =

a11 . . . a1n...

. . ....

am1 . . . amn

v1...vn

=

a11v1 + a12v2 + · · ·+ a1nvna21v1 + a22v2 + · · ·+ a2nvn

...am1v1 + am2v2 + · · ·+ amnvn

∈ Rm .

We have: (m×n-matrix) · (vector of size n) = (vector of size m).

Proposition 2.4. We have for A ∈ Rm×n, x, y ∈ Rn and λ ∈ R

i) A(x+ y) = Ax+Ay,

ii) A(λx) = λ(Ax).

Definition 2.5. For a matrix A = (aij) ∈ Rm×n and a vector b ∈ Rm the matrix

(A | b) =

a11 . . . a1n b1...

. . ....

...am1 . . . amn bm

∈ Rm×(n+1)

is called the augmented matrix of the linear system Ax = b.

The augmented matrix (A | b) is just the matrix A where we append the vector b as a column. Theline | is a useful notation to distinguish between the left- and right-hand side of the correspondinglinear system but it has no mathematical meaning. We will view (A | b) as a usual matrix with mrows and n+ 1 columns.

Definition 2.6. The following operations on a matrix are called elementary row operations.

(R1) Add a multiple of one row to another row.

(R2) Multiply a row with a non-zero number.

Version 1 (October 3, 2021)- 3 -

Tutorial exercise

34 CHAPTER 1 Linear Equations

Solving the linear system A!x = !b amounts to expressing vector !b as a linearcombination of the column vectors of matrix A.

EXAMPLE 15 Write the system ∣∣∣∣2x1 − 3x2 + 5x3 = 79x1 + 4x2 − 6x3 = 8

∣∣∣∣in matrix form.

Solution

The coefficient matrix is A =[

2 −3 59 4 −6

], and !b =

[78

]. The matrix form is

A!x = !b, or[

2 −3 59 4 −6

]

x1

x2

x3

=[

78

].

!Now that we can write a linear system as a single equation, A!x = !b, rather than

a list of simultaneous equations, we can think about it in new ways.For example, if we have an equation ax = b of numbers, we can divide both

sides by a to find the solution x :

x = ba

= a−1b (if a "= 0).

It is natural to ask whether we can take an analogous approach in the case of theequation A!x = !b. Can we “divide by A,” in some sense, and write

!x =!bA

= A−1!b?

This issue of the invertibility of a matrix will be one of the main themes ofChapter 2.

EXERCISES 1.3GOAL Use the reduced row-echelon form of the aug-mented matrix to find the number of solutions of a linearsystem. Apply the definition of the rank of a matrix. Com-pute the product A!x in terms of the rows or the columnsof A. Represent a linear system in vector or matrixform.

1. The reduced row-echelon forms of the augmentedmatrices of three systems are given here. How many so-lutions does each system have?

a.

1 0 2 00 1 3 00 0 0 1

b.[

1 0 50 1 6

]

c.[

0 1 0 20 0 1 3

]

Find the rank of the matrices in Exercises 2 through 4.

2.

1 2 30 1 20 0 1

3.

1 1 11 1 11 1 1

4.

1 4 72 5 83 6 9

5. a. Write the system∣∣∣∣

x + 2y = 73x + y = 11

∣∣∣∣

in vector form.b. Use your answer in part (a) to represent the system

geometrically. Solve the system and represent thesolution geometrically.

6. Consider the vectors !v1, !v2, !v3 in R2 (sketched in theaccompanying figure). Vectors !v1 and !v2 are parallel.How many solutions x , y does the system

x !v1 + y !v2 = !v3

have? Argue geometrically.

v#3

v#1

v#2

Exercises from the reference book

1.3 On the Solutions of Linear Systems; Matrix Algebra 35

7. Consider the vectors !v1, !v2, !v3 in R2 shown in the ac-companying sketch. How many solutions x , y does thesystem

x !v1 + y !v2 = !v3

have? Argue geometrically.

v!3

v!1

v!2

8. Consider the vectors !v1, !v2, !v3, !v4 in R2 shown in theaccompanying sketch. Arguing geometrically, find twosolutions x , y, z of the linear system

x !v1 + y !v2 + z !v3 = !v4.

How do you know that this system has, in fact, infinitelymany solutions?

v!3

v!1

v!2v!4

9. Write the system∣∣∣∣∣∣

x + 2y + 3z = 14x + 5y + 6z = 47x + 8y + 9z = 9

∣∣∣∣∣∣

in matrix form.

Compute the dot products in Exercises 10 through 12(if the products are defined).

10.

123

·

1

−21

11.[1 9 9 7

]·

666

12.[1 2 3 4

]·

5678

Compute the products A!x in Exercises 13 through 15 us-ing paper and pencil. In each case, compute the producttwo ways: in terms of the columns of A (Theorem 1.3.8)and in terms of the rows of A (Definition 1.3.7).

13.[

1 23 4

] [7

11

]14.

[1 2 32 3 4

]

−1

21

15.[1 2 3 4

]

5678

Compute the products A!x in Exercises 16 through 19using paper and pencil (if the products are defined).

16.[

0 13 2

] [2

−3

]17.

[1 2 34 5 6

] [78

]

18.

1 23 45 6

[

12

]19.

1 1 −1

−5 1 11 −5 3

123

20. a. Find

2 34 56 7

+

7 53 10 −1

.

b. Find 9[

1 −1 23 4 5

].

21. Use technology to compute the product

1 7 8 91 2 9 11 5 1 51 6 4 8

1956

.

22. Consider a linear system of three equations with threeunknowns. We are told that the system has a unique so-lution. What does the reduced row-echelon form of thecoefficient matrix of this system look like? Explain youranswer.

23. Consider a linear system of four equations with threeunknowns. We are told that the system has a unique so-lution. What does the reduced row-echelon form of thecoefficient matrix of this system look like? Explain youranswer.

24. Let A be a 4 × 4 matrix, and let !b and !c be two vec-tors in R4. We are told that the system A!x = !b has aunique solution. What can you say about the number ofsolutions of the system A!x = !c?

25. Let A be a 4×4 matrix, and let !b and !c be two vectors inR4. We are told that the system A!x = !b is inconsistent.What can you say about the number of solutions of thesystem A!x = !c?

26. Let A be a 4 × 3 matrix, and let !b and !c be two vec-tors in R4. We are told that the system A!x = !b has aunique solution. What can you say about the number ofsolutions of the system A!x = !c?

27. If the rank of a 4 × 4 matrix A is 4, what is rref(A)?

28. If the rank of a 5 × 3 matrix A is 3, what is rref(A)?

In Problems 29 through 32, let !x =

53

−9

and !y =

201

.

29. Find a diagonal matrix A such that A!x = !y.

30. Find a matrix A of rank 1 such that A!x = !y.

31. Find an upper triangular matrix A such that A!x = !y,

L

III

Linear Algebra I & Mathematics Tutorial 1bNagoya University, G30 Program

Fall 2021Instructor: Henrik Bachmann

Homework 2: Matrices, vectors & the rank of a matrix

•

Deadline: 31th October, 2021

This homework is not online yet. It will come online in the coming days.

Exercise 1. (3+3 = 6 Points) Show that for all A ∈ Rm×n, x, y ∈ Rn and λ ∈ R we have

i) A(x+ y) = Ax+Ay,

ii) A(λx) = λ(Ax).

(Without using Proposition 2.4. from the lecture).

Exercise 2. (4 Points) Let p(x) = a0+a1x+a2x2+a3x3 be a polynomial of degree 3 with real coefficientsa0, a1, a2, a3 ∈ R. For this polynomial p we define the vector vp by

vp =

a0a1a2a3

∈ R4 .

Find a matrix D ∈ R4×4, such that vp′ = Dvp, where p′ denotes the derivative of the polynomial p withrespect to x. What is the rank of D?

Exercise 3. (4+3+1 = 8 Points) Let a, b, c, d ∈ R and A =

(a bc d

).

i) Show that rk(A) = 2 if and only if ad− bc #= 0.

ii) What can you say about a, b, c, d if rk(A) = 1? Consider the following subset of R2

L = {x ∈ R2 | x = Av for some v ∈ R2}.

How does L look like if rk(A) = 1? How does it look like if if rk(A) = 2?

iii) What can you say about a, b, c, d if rk(A) = 0?

Version: October 12, 2021- 1 -

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