Linear Algebra Answers

Let V be the set of all vectors of the form (x1, x2, x3) in R

3

(a) x1 − 3x2 + 2x3 = 0.

(b) 3x1 − 2x2 + x3 = 0 and 4x1 + 5x2 = 0.

Find the dimension and basis for V.

 Let f : A → B be a one-to-one correspondence.

1. Prove that f^-1 is a function.

2. Prove that f^-1 is one-to-one.

3. Prove that f^-1 is onto.

4. Conclude that f^-1 : B → A is a one-to-one correspondence.

For each of the following functions determine the image of S = {x ∈ R : 9 ≤ x²}.

1. f : R → R defined by f(x) = |x|.

2. g : R → R⁺ defined by g(x) = e^x.

3. h : R → R defined by h(x) = x − 9

The Nature of quadratic form Q(x) = x² + 3y² +3z² is 

9(b) Consider the system of equations (5)

11 2x1 + x2 + 4x3 =

3x1 + x2 + 5x3 =14

A feasible solution is x1 = ,2 x2 = ,3 x3 = .1 Reduce this feasible solution to a basic

feasible solution.

9. (a) Show that the set {( , 3|) 5 15}

2 2

S = x y x + y ≤ is convex.

8. (a) A manufacturer has two products P1

 and , P2

 both of which are produced in two steps 

by machines M1

 and . M2

 The process time per hundred for the products on the 

machines

M1 M2

Profit (in 

thousand Rs. 

per 100 units) 

P1

4 5 10 

P2

5 2 5 

Available 

hours 

100 80 

 The manufacturer can sell as much as he can produce of both products. Formulate the 

problem as LP model. Determine optimum solution, u

7(b) Is the set of vectors {( ),3,2,1 ),1,4,3( )}2,3,2( linearly independent? Give reasons for 

the answer. (3) 

6 4 1 5 14 

8 9 2 7 16 

4 3 6 2 5 

6 10 15 4

6(b) Write the dual of the following LPP after reducing it to canonical form. (4) 

 Min 3 1 4 2 3 3 Z = x + x + x

 Subject to 

 2x1 + 4x2 =12

 11 5x1 + 3x3 ≥

 6x1 + x2 ≥ 8

 x1

, x2

, x3 ≥ 0

Three water purification facilities can handle at most 10 million gallons in a certain time period. Plant 1 leaves 20% of certain impurities, and costs P20,000 per million gallons. Plant 2 leaves 15% of these impurities and costs P30,000 per million gallons. Plant 3 leaves 10% impurities and costs P40,000 per million gallons. The desired level of impurities in the water from all three plants is at most 15%. If Plant 1 and Plant 3 combined must handle at least 6 million gallons, find the number of gallons each plant should handle so as to achieve the desired level of purity at minimum cost.