Let L be a linear operator on Rn Suppose thatLx 0 for some

Let L be a linear operator on Rⁿ. Suppose thatL(x) = 0 for some x that does not = 0. Let A be the matrix representing L with respect to the standard basis[e₁, e_2, .....e_n]. Show that A is singular.

Solution

given that for some nonzero vector x, L(x) = O

so, the set of n vectors comprising this x willhave the image

vectors comprising zero vector and so the n*n matrix relativeto L will contain one zero column. so, det([L,B]) =0.

so,[L,B] is a singular matrix.