If I have a system of equations:
a1x+b1y+c1z=d1a1x+b1y+c1z=d1
a2x+b2y+c2z=d2a2x+b2y+c2z=d2
a3x+b3y+c3z=d3a3x+b3y+c3z=d3
where the coefficients ai,bi,ciai,bi,ci and constants didi are real, then I know that a zero determinant on the coefficient matrix tells us that we have an inconsistent system, and there are either infinitely many solutions or zero solutions. I read that if the column vector (d1,d2,d3)T(d1,d2,d3)T lies outside of the "column space" of our coefficient matrix then we have zero solutions (makes sense as the matrix cannot map to this vector). My question is, e.g., for the matrix
[523121322]⎡⎣⎢513222312⎤⎦⎥
how can we check if it maps to e.g. [421]⎡⎣⎢421⎤⎦⎥
using this column space property? (i.e. what are the actual steps involved?)
Also, can anyone clarify what exactly is meant by the column space?
