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## Homework Statement

Show that {(1, 2, 3), (3, 4, 5), (4, 5, 6)} does not span R

^{3}. Show that it spans the subspace of R

^{3}consisting of all vectors lying in the plane with the equation x - 2y + z = 0.

## Homework Equations

## The Attempt at a Solution

I made a matrix of:

A = [ 1 3 4 ; 2 4 5; 3 5 6] and reduced it to row-echelon form to determine rank. I found that rank(A) = 2. Therefore, it can't span R

^{3}, since rank(A) < n for R

^{n}.

But then I need to show that it spans the plane given by the equation x - 2y + z = 0.

I thought that perhaps rewriting the problem as: z = -x + 2y and plugging back in for z. But that doesn't really tell me anything.

I also tried to show the span by demonstrating where it has linear combinations by writing span(F) = span{ax - 2by + cz = 0} but I don't know what to do from here.

I was able to find the solution in the solutions manual, but I can't decipher what it's saying or how it did it. It's not using a method described in the book.

Could someone explain this better or give me another method to figure out whether a set spans a subspace?