Question

(b) Find a linear transformation T: R3 → R4, whose null space is generated by (0, 1, -3) and (0, -3, 4).​

Answer 1

Answer:

is an m X n matrix. Check the true statements below: The null space of A is the solution set of the equation Ax = 0. If the equation Ax = b is consistent, then ColA is R^m. The column space of A is the range of the mapping x rightarrow Ax. The null space of an m X n matrix is in R^m. ColA is the set of all vectors that can be written as Ax for some x The kernel of a linear transformation is a vector space.

Answer 2

Answer:

T(x, y, z) = {( 0, y, z) : y, z ∈ R}

Step-by-step explanation:

We need to find the linear transformation T: R³⇒R⁴ whose null space is generated by (0, 1, -3) and (0, -3, 4).

Let {(1, 0, 0), (0, 1, 0). (0, 0, 1)} be the basis of T.

So, the given pairs of vectors will generate the subspace {( 0, y, z) : y, z ∈ R} since they are linearly independent.

By Rank-Nullity theorem, the dimension of image space in R⁴ is of dimension 3-2=1.