The linear transformation T is represented by T(v)=Av.Find a basis for (a) the kernel of T and (b) the range of T:-

Question

A= ⌈ 1 2⌉

⌊ 3 4 ⌋

Answer 1

a) To get kernel we solve system Ax = 0. Let do this swiching matrix to rref: [1 2 0] [1 2 0]

[3 4 0] -3R₁ + R₂ → R₂ [0 -2 0]

Then we add second row to 1st and reduce 2nd row by -2 we get [1 0 0]

[0 1 0].

So we have only zero solution x = 0 and y = 0. Kernel of k{T} = (0,0) only zero-vector.

Basis of k(T) only (0,0) vector.

b) Dimension of k(T) is 1(one vector in basis). Range of kernel is 1