1. The problem statement, all variables and given/known data Let B be 4x4 matrix: [ 1 3 -2 2 ] [ 0 1 1 -5 ] [ 1 2 -3 7 ] [ -2 -8 2 -1 ] a) Do the columns of B span R^4? b) Does the equation Bx = y have a solution for each y in R^4? Sorry for the crappy matrix, its a 4x4, R^4 is the funky double R for 'Reals' 2. Relevant equations A) So, started out with row reduction. To find B spans R^4, B must have pivot positions in every row. I won't go through all the row reduction steps, but I came out to a matrix looking like so: [ 1 3 -2 2 ] [ 0 1 1 -5 ] [ 0 0 0 0 ] [ 0 0 0 -7 ] There are only pivot positions in row 1 + 2, so no, B does NOT span R^4. B) Bx = y does NOT have solutions for each y in R^4. This is due to (what we called in class Theorem 4) with the logically equivalent assumptions that: 1) For each y in R^4, Bx = y has solution 2) Each y (element of R^4) is linear combo of A columns 3) Columns of A span R^4 4) B has pivot in every row Since B does not span R^4 and does not have pivots in every row, hence forth the statement that Bx = y has solutions for y in R^4 is incorrect. ...Does this look right to anyone?