(1) Define T:R->R be a linear transformation such that T(x,y,z)= (2x,2y,2z) then the given value of T is
A. 3
B. 2
C. 4
D. 6
(A) (B) (C) (D)
(2) Let V and W be vector spaces over a field F, and let T:V-> W be a linear transformation then only one of the following statement is correct
(A) V=R(T)
(B) Ker T/V=R(T)
(C)V/KerT=R(T)
(D) R(T)=DIM(T)
(A) (B) (C) (D)
(1) To find the value of T, we need to substitute the given input into the linear transformation equation T(x,y,z) = (2x, 2y, 2z).
Given T(x, y, z) = (2x, 2y, 2z), we can substitute any values for x, y, and z, and follow the transformation to find the resulting output. In this case, we can substitute x = 1, y = 1, and z = 1 to get:
T(1, 1, 1) = (2(1), 2(1), 2(1)) = (2, 2, 2)
Therefore, the value of T is (2, 2, 2).
Now, let's match this value against the given options to determine the correct answer:
(A) 3 - This is not the correct value.
(B) 2 - This is the correct value.
(C) 4 - This is not the correct value.
(D) 6 - This is not the correct value.
Therefore, the correct answer is B. 2.
(2) The given options are:
(A) V = R(T)
(B) Ker T/V = R(T)
(C) V/Ker T = R(T)
(D) R(T) = DIM(T)
Let's evaluate each option to determine which one is true:
(A) V = R(T) - This option states that the vector space V is equal to the range of the linear transformation T. This is not necessarily true, as V could be greater than or equal to the range of T.
(B) Ker T/V = R(T) - This option states that the kernel of T divided by V equals the range of T. This is not true in general, as the kernel of T is a subspace of V and cannot be directly divided by V.
(C) V/Ker T = R(T) - This option states that the quotient space V divided by the kernel of T equals the range of T. This is actually a true statement known as the First Isomorphism Theorem for vector spaces. Therefore, this option is correct.
(D) R(T) = DIM(T) - This option states that the range of T is equal to the dimension of T. This is not true in general, as the range of T can have a dimension smaller than the dimension of T.
Therefore, the correct statement is C. V/Ker T = R(T).