Linear Algebra Answers

let T:P2-->P2 defined by T(a+bx+cx^2)=(2b+3cx+(a-b)x^2) check T is linear trnsfrmtn.find matrix wrt bases B={x2,x2+x,x2+x+1} ,B2={1,x,x2}.find kernel of T

A car rental company owns five types of cars: Small,Economy, Standard, Family, Exclusive. The prices for per day rent for eachtype of the cars are given in British pounds by the1×5matrix

Suppose u and v are nonzero vectors in 3-space, where u=(u1, u2, u3) and v=(v1, v2, v3). Prove that u x v is perpendicular to both u and v by making use of the dot product.

Suppose | a b c |

| d e f | =8

| g h i |


Find value of | (g+2a) (h+2b) (i + 2c) |

| 3a 3b 3c |

| 2d 2e 2f |


I got 48 as my answer. please confirm if correct.

Suppose | a b c |

| d e f | = 8

| g h i |


Find value of | ( g +2a) (h+ 2b) (i + 2c) |

| 3a 3b 3c |

| 2a 2b 2c |

I got 48, please confirm if this is the correct answer or not.

Find a basis for R(A)^⊥, where R(A) denotes the row space of the matrix

A where A is a 3x4 matrix (1 0 4 0 \ 0 1 2 -1 \ 1 -1 2 1)

Find the dual basis for the basis {1,1+x,x²-1} of the vector space P3={a0+a1x+a2x²:a0,a1,a2 belongs to R}

Show that if S and T are linear

transformations on a finite dimensional

vector space, then rank (ST)<= rank (T). Also

give examples of linear transformations S

and T for which rank (ST) <rank (T).

Find the range space and the kernel of the

linear transformation :

T: R4 --> R4, T(x1, x2, x3, x4) =

(x1+ x2+ x3+ x4, x1+ x2, x3+ x4, 0)

Let T : R3-> R3be the linear operator

defined by T(x1, x2, x3) = (x1, x3, -2x2- x3).

Let f(x) = - x³+ 2. Find the operator f(T).