Linear Algebra Answers

let T be a linear map on R³ Such that T(1,0,0)=(1,1,1),T(0,1,0)=(0,3,5),T(0,0,1)=(2,2,2)

a)find matrix A representing T with respect to the usual basis of R³

b)find matrix B representing T with respect to the basis S={(1,2,3),(2,3,4),(-1,0,1)}

reduce the following quadratic form 3x²-2y²-z²+12yz-4xy+8zx to canonical form by an orthogonal transformation. Also find the rank, index signature and nature

solve the following question.

[ 2 6 7 ] [ -12 ]

[ -1 4 8 ] [ 6 ]

[ 3 -2 -5 ] [ -8 ]

a. R2 <=> R3

b. -1\2 R3

c. -4 R2 + R1 -> R1

d. -1\2 R1 + R2 -> R2

solve a system step-by-step of linear equations using Gaussian elimination.

2x + 7y + z = 1

x + 3y - z = 2

x + 7y + 12z = 45

Find the row echelon form of [

1 2 − 2 − 2 −3

1 3 − 2 0 −4

3 8 − 7 − 2 −11

2 1 − 9 − 10 − 3

]. 

Determine whether the matrix (

1/√3 1/√6 −1/√2

1/√3 −2/√6 0

1/√3 1/√6 1/√2

) is Orthogonal or not.

Solve the following system of linear equations by the elementary row operations: 5

𝑝 + 2𝑞 − 2𝑟 − 𝑡 = 0

2𝑝 + 5𝑞 − 3𝑟 − 𝑡 = 1

3𝑝 + 8𝑞 − 4𝑟 − 𝑡 = 2

𝑝 + 5𝑞 + 𝑟 + 2𝑡 = 3

If On denotes the vector space of all polynomial of degree ≤ n , give two linearly independent elements of P4/PE.

Find the dual basis of {(1,0,1),(1,1,0),(0,1,1)} in R^3

In a study of the domestic market share of three major automobile manufacturers A, B and C in a

certain country, it was found out that of the customers who bought a car manufactured by A, 75%

would again buy a car manufactured by A, 15% would buy a car manufactured by B and the rest

wouldbuyacarmanufacturedbyC.OfthecustomerswhoboughtacarmanufacturedbyB,90%

would again buy a car manufactured by B, 5% would buy a car manufactured by A and the rest

would buy a car manufactured by C. Of the customers who bought a car manufactured by C, 85%

would again buy a car manufactured by C, 5% would buy a car manufactured by A and the rest

would buy a car manufactured by B.

Required

The long run market share of the manufacturers (14 Marks)