Question

Hi,Is there anyone do the assignment5,problem33?I think there are some mistake in the answer

Answer 1

I think r1 and r2 must be linear! Or ,rank will become 2 not 1.then,there only will be zero in the N(A)space.

Answer 2

You're correct. If we have a 2 by 2 matrix A and r₁, r₂ are a basis for C(Aᵀ) then N(A) must be the zero vector.

The mistake is actually in the question not the answer. In the book the question is:

Suppose I give you eight vectors r₁, r₂, n₁, n₂, c₁, c₂, l₁, l₂ in R⁴.

a) What are the conditions for those pairs to be bases for the four fundamental subspaces of a 4 by 4 matrix?
b) What is a possible matrix A?

The answer is correct for that question as far as I'm aware.