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Determine the real root of 𝑓(𝑥)=𝑥4−8𝑥3−35𝑥2+450𝑥 −1001. Using false-position method to locate the root. Employ initial guesses of 𝑥l=4.5 and 𝑥u=6 and iterate until the estimated error 𝜀a falls below the level of 𝜀s=1.0%

Assume T and S are matrices of the same size, Prove or disprove that (T+S)2 is symmetric , skew symmetric or neither

Find an expression for a square matrix A satisfying A 2 = In, where In is the n × n identity matrix. Give 3 examples for the case n = 3.

A set S of vectors in R4 is given. Find a subset of S that forms a basis for the subspace of R4 spanned by S.

V1= (3,3,-6,9) V2=(2,30,-60,38) v3=(4,32,-64,44)

A basis for the subspace is given by___?

Show that in a finite dimensional vector space V(F) whose basic set is B={x₁,x₂,...xₙ} every vector x∈V is uniquely expressible as linear combination of the vector in B.

In the vector space Vs(Rₙ) let α=(1,2,1), β=(3,1,5), -γ=(3,-4,7) prove that the sub space planned by S={α,β} and T={α,β,γ} are same.

If p(x) denotes the set of all polynomials one indeterminates x over field F, then show that p(x) is a vector space over F with addition defined as addition of polynomials and scalar multiplication defined as the product of polynomials by an element of F. i.e if p(x)={p(x)/p(x)=a₀+a₁x+...+aₙxₙ...}={∑∞,ₙ₌∞ aₙxⁿ for as ∈ f}.

Define addition and scalar multiplication to prove.

Let V be set of real valued continuous function defined as [0,1] such that f(0/3)=2. Show that V is not a vector space over R (reals) under addition and scalar multiplication defined as : (f+g)(x)=f(x)+g(x) for all f,g € V.

(alpha f)(x)= alpha f(x) for all alpha € R, f€V.

Given the following quadratic form involving three variables,

Q (x1, x2, x3) = 5x21 + 8x1x3 + 3x22 - 6x2x3 + 10x23

a. Derive the symmetric matrix associated with Q

b. Determine the definiteness of the matrix you derived in a

A company makes products A and B. Each unit of product A requires 1 unit of resource 1 and 5 units

of resource 2. Each unit of product B requires 5 units of resource 1 and 4 units of resource 2. If there

are 40 units of resource 1 and 140 units of resource 2 available, how many units of each product should

be produced if all the resources are to be used?

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