Linear Algebra Answers

Consider the vector space P₃

_5.1. Is span {1+x,x+x²,x²+x³,x³+1}=P₃ Motivate your answer

5.2. Let D P₃→ P₂ be the differentiation operator D(a₀+a₁x+a₂x²+a₃x³)=a1+2a₂x+3a₃x²

(i) Find the matrix representation of D relative to the basis {1,x,x²,x³} using the coefficient ordering a0+a₁x+a₂x²+a₃x³→"\\begin{bmatrix}\n a0 \\\\\n a1\\\\\n a2 \\\\\n a3\n\\end{bmatrix}"

(ii) Find the kernel and range of D

Consider the matrix A="\\begin{bmatrix}\n 1 & 0 & 2 \\\\\n 0 & 1 & 0 \\\\\n 2 & 0 & 1\n\\end{bmatrix}"

1.1. Show that the characteristic equation for the eigenvalues "\\lambda" of A is given by (λ²-1) (λ-3)=0.

1.2. Find an orthogonal matrix P which diagonalizes A.

1.3. Find Aⁿ (for n ∈ N) as a matrix

Express M as a linear combination of the matrices A, B, C, where M = [

4 7

7 9

] , A = [

1 1

1 1

] , B

= [

1 2

3 4

] , C = [

1 1

4 5

] .

Show that the vectors u1 = (1, 1, 1), u2 = (1, 2, 3), u3 = (1, 5, 8) span R

3

.

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, Sh30 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 brand

A 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.

Show that P^-1HP = 3à —3 matrix

Use cofactor method to find the inverse of P

consider the vector space of R³

2.1. Show that <x, y>=x₁y₁+2x₂y₂+x₃y₃, x="\\begin{bmatrix}\n x1 \\\\\n x2 \\\\\nx3\n\\end{bmatrix}" ,y="\\begin{bmatrix}\n y1 \\\\\n y2 \\\\\n y3\n\\end{bmatrix}""\\in" R³

^ 2.2. Are the vectors "\\begin{bmatrix}\n 1 \\\\\n 1 \\\\\n 1\n\\end{bmatrix}", "\\begin{bmatrix}\n 1 \\\\\n 1 \\\\\n - 1\n\\end{bmatrix}", "\\begin{bmatrix}\n 1 \\\\\n -1 \\\\\n -1\n\\end{bmatrix}" Linearly independent?

2.3. Apply the Grem-Schmidt process to the following subset of R³ "\\begin{bmatrix}\n 1 \\\\\n 1 \\\\\n 1\n\\end{bmatrix}", "\\begin{bmatrix}\n 1 \\\\\n 1 \\\\\n - 1\n\\end{bmatrix}", "\\begin{bmatrix}\n 1 \\\\\n -1 \\\\\n -1\n\\end{bmatrix}" to find an orthogonal basis with the inner product defined in 2.1. for the span of this subset

Reduce the quadratic form

2 2 2 8 7 3 12 – 8 4 x y z xy yz zx    

to the canonical form

through an orthogonal transformation and hence show that it

is positive Semi-definite.

let n€N consider the set of nxn symmetric matrices over R with the usual addition and multiplication by a scalar.

a) Show that this set with the given operations is a vector subspace of M_nn

_{b) What is the dimension of this vector subspace?}

_{c) Find a basis for the vector space of 2x2 symmetric matrices.}

let n€N consider the set of nxn symmetric matrices over R with the usual addition and multiplication by a scalar. show that this set with the given operations is a vector subspace of M_nn