Linear Algebra Answers

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

Suppose 𝐴 and 𝑃 are 3 × 3 matrices and 𝑃^-1 exists. If 𝑃^-1𝐴𝑃 = ( 1 2 3

0 4 5

0 0 6 ) then find all eigen values of 𝐴² .

If a third degree polynomial has a lone x-intercept at x=a, discuss what this implies about the linear and quadratic factors of that polynomial.