Question

Define Rank-Nullity Theorem and verify same for linear transformation T: R3 → R 3 defined by T(x, y,z)  (x  2y, y  z, x  2z)​

Answer 1

Answer:

The nullity of a linear transformation is the dimension of the kernel, written nulL=dimkerL. Let L:V→W be a linear transformation, with V a finite-dimensional vector space.

Answer 2

Answer:

More precisely, the matrix

(12−2111)

is the matrix associated to T with respect to the standard bases of R3 and R2. Without doing reduction, the rank of T is given by the rank of one of the biggest submatrices with non-vanishing determinant. In your case there is a submatrix of rank 2 with determinant non-zero (as gimusi is showing), so the rank of T is 2.