Linear Algebra Answers

Let V be the set R⁺ of positive real number and define V₁ "\\bigoplus" V₂=V₁V₂ and "\\delta" "\\bigodot" V₁=V₁"\\delta" for all V₁V₂"\\epsilon" V and "\\delta" "\\epsilon" R. Then show that v is a vector space over R

a company in the wholesale trade selling sportswear and stocks two brands, a and b, of football kit, each consisting of a shirt, a pair of shorts and a pair of socks. the costs for brand a are sh.50 for a shirt, sh 30 for a pair of shorts and sh10 for a pair of socks and those for brand b are sh.60.25 for a shirt, sh.40 for a pair of shorts and sh.10 for a pair of socks. three customers x, y, z demand the following combinations of brands; x, 36 kits of brand a and 48 kits of brand b y, 24 kits of brand a and 72 kits of brand b z, 60 kits of branda required:i) express the costs of brands a and b in matrix form, then the demands of the customers, x,y and z in matrix form. (5 marks)ii) by forming the product of the two matrices that you obtain in the previous part, deduce the detailed costs to each of the customers. (5 marks)

Solve the system of equations 2x + y + 5z = 18 5x + 3y – 2z = 2 x – 6y + 2z = 1 using Gauss elimination method with partial pivoting. Show all the steps.

Consider the linear eigenproblem Ax=λx for the matrix

D=[■8(1&1&2@2&1&1@1&1&3)]

1.Solve for the largest (in magnitude) eigenvalue of the matrix and the corresponding eigenvector by the power method with x^(0)T=[■8(1&0&0)].

2.Solve for the smallest eigenvalue of the matrix and the corresponding eigenvector by the inverse power method using the matrix inverse. Use Gauss-Jordan elimination to find the matrix inverse.

Which of the following functions has one-to-one correspondence:

(i) f: R->R defined as f(x) = 3x + 2

(ii) g: R -> defined as g(x) = X2 - 1

2. The square root of (2 - i) is...

3. Suppose lambda E C such that lambda(2+i, 3 - i) = (3 - i, 4 - 4i). Then...

(i) Lambda = (1 + i)

(ii) lambda = (2 - i)

(iii) lambda Does not exist

4. For a,b E R, let S be a subset of R2 defined as S = { (x,y) E R3 : x+y + axy = b }. Then S is a subspace of R2 if...

5. Suppose U = {(x,y,x+y,z,2y+z) E F5 : x, y, z E F }. Then a subspace W of F5 such that F5 = U O+ (plus sign inside the circle) W is ...

(i) W = { (0,0,a,b,c) E F5 : a,b,c E F}

(ii) W = { (0,0,a,0,b) E F5 : a, b E F}

6.. For a given function f: R->R defined as f(x) = 2x - 1, the image of set S = { x E R: x2 - 4 >= 0 } is...

1. Let S be the subspace of R⁵ defined by S = { (x₁, x₂, x₃, x₄, x₅) E R⁵ : x₁ = x_2, x₃ = 2x₄ + x₅ }. Then the dimension of S is
2. Let T: R³ -> R³ be defined as T (x,y,z) = (x+y, x-y, x+2z). Then the basis of range T is...
3. Which of the following transformations are linear:

(i) T₁ : R³ -> R² by T₁ (u, v, w) = ( u - v + 2w, 5v - w).

(ii) T₂ : P (R) -> R by T₂ (P) = (integral sign from b to a) 2p(x)dx for a,b E R with a<= b

(iii) T₃ : P(R) -> P(R) by T₃ (P(U) = UP (U) + U

4.. Suppose T : R² -> M₂₂ is a linear defined by T (u,v) = [u v]

[u 2u]

then Ker (T) is...

5.. Suppose T: R⁶ -> R⁴ is a linear map such that null T = U where U is a 2-dimentional subspace of R⁶ . Then dim range T is...

6.. For a given 2x2 matrix A = [ 5 -3 ]

[ -6 2]

the matrix P that is diagonalizes A can be written as P = ...

what is the square root of 2-i?

1. Let S be the subspace of R⁵ defined by S = { (x₁, x₂, x₃, x₄, x₅) E R⁵ : x₁ = x_2, x₃ = 2x₄ + x₅ }. Then the dimension of S is
2. Let T: R³ -> R³ be defined as T (x,y,z) = (x+y, x-y, x+2z). Then the basis of range T is...
3. Which of the following transformations are linear:

(i) T₁ : R³ -> R² by T₁ (u, v, w) = ( u - v + 2w, 5v - w).

(ii) T₂ : P (R) -> R by T₂ (P) = (integral sign from b to a) 2p(x)dx for a,b E R with a<= b

(iii) T₃ : P(R) -> P(R) by T₃ (P(U) = UP (U) + U

Consider the linear eigenproblem Ax=λx for the matrix

D=[1 1 2

2 1 1

1 1 3 ]

1. Solve for the largest (in magnitude) eigenvalue of the matrix and the corresponding eigenvector by the power method with 𝑥^(0)T=[1 0 0]

2. Solve for the smallest eigenvalue of the matrix and the corresponding eigenvector by the inverse power method using the matrix inverse. Use Gauss-Jordan elimination to find the matrix inverse.