Linear Algebra Answers

Are the following vectors linearly independent?

1. (1, 2), (-2, 4)
2. (0, 0, 1), (0, 1, 1), (1, 1, 1)
3. (1, 2), (1, 3), (1, 1)
4. (1, 2, 3), (2, 3, 4)
5. (0, 1, 2), (2, 0, 1), (0, 0, 1), (3, 2, 1)

Use Cayley hamilton theorem to find the values of the matrix

A^8-5A^7+7A^6-3A^5+8A^4-5A^3+8A^2-2A+I

Q1: Use the matrix

9 1

7 2 

(a)  Obtain the Hill cipher for the plaintext message

                                              PAK ARMY

by letting  A=1,B=2,C=3,.......,Y=25 and Z=26

(b)  Decode Hill 2- cipher which was encrypted by this matrix.

Given the linear transformation below:

T (x₁, x₂, x₃) → (x₁-x₂+2x₃, 2x₁-2x₃, -x₁-x₂+4x₃, 3x₁-x₂)

T = R³ → R⁴

1. Determine the transformation matrix of the linear transformation above

2. Determine Ker(T) and Range(T)

It is known that the vectors in the vector space R³:

⃗v₁ = (1, 1, 1), v₂ = (2, -1, 1), v₃ = (0, 2, 1)

and

⃗w₁ = (2, -1, 3), w₂ = (3, -1, 7), w₃ = (-1, 1, 1)

The vectors v₁, v₂, v₃ are basis in R³. Transformation T : R³ → R³ is a linear transformation defined by:

 T( ⃗v_i) = ⃗w_i

Define:

1. Matrix transformation of T

2. Basis of Ker(T) and Range(T)

If the cofactor matrix is

7 15 9

6 74

10 2 7

−

 

 

−

 

 

−

 

, then the Adjoint matrix is

Determine if the following transformation is a linear transformation:

T (x₁, x₂) → (cos(x₁), sin(x₂))

T = R²→R²

Question 4 Use row operation to show that

det T = 0









x

2 2x + 1 4x + 4 6x + 9

y

2 2y + 1 4y + 4 6y + 9

z

2 2z + 1 4z + 4 6z + 9

w

2 2w + 1 4w + 4 6w + 9









If a stock price goes from $10 to $12 from January 1st to January 31, from $12 to $9 from February 1st to February 28th, and from $9 to $15 from March 1st to March 31th is the price change from $10 to $15 a straight line?

It is clear that in each of the three time intervals mentioned there was a complex daily variation of prices as in an electrocardiogram. But what would be a simplified solution for a first naive view of the situation? Would a simple function hold up? What is the simplest function to represent this situation? Does your naïve initial and simplified model allow you to predict the behavior of the stock in the next month?

How can I use three “pieces” of lines to describe the price movements from the beginning of January to the end of March? Show the graph for the price movement.

(1) The table below shows the calories, fat, and carbohydrates per ounce for three brands of cereal.

Calories Fat Carbohydrates

Cereal Brand A 10 0 11

Cereal Brand B 100 0.2 22.5

Cereal Brand C 130 5.6 19

Total Required 400 6.2 92

(i) Using the information in the table above, write a system of three equations.

(ii) Write a system in matrix form Ax = b.

(iii) Using the Cramer's Rule or Inverse Method, find the amount of each brand of cereal that will give the level of nutrition required.