Linear Algebra Answers

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).

1.Let B = (a1,a2, a3) be an ordered basis of
R3 with al= (1, 0, -1), a2= (1, 1, 1),
a3= (1, 0, 0). Write the vector v = (a, b, c) as
a linear combination of the basis vectors
from B.

2.Suppose al= (1, 0, 1), a2= (0, 1, -2) and
a3 = (-1, -1, 0) are vectors in R3and
f : R3 -> R is a linear functional such that
f(al) = 1, f(a2) = -1 and f(a3) = 3. If
a = (a, b, c) E R3, find f(a).

Given the basis {(1, - 1, 3), (0, 1, - 1),
(0, 3, - 2)} of R3, determine its dual basis.

A is a 3x4 matrix where A= (1 1 0 0 \ -1 3 0 1 \ -3 1 -2 1)
Find an orthonormal basis for the row space of the matrix using the Gram-Schmidt Process