In the book Linear algebra and it’s application by Gilbert Strang, it is given that
We have an augmented matrix for system of linear equations Ax=b as:
After applying gaussian elimination:
Description of Column space of A: The column space contains all vectors with b3+b2−5b1=0.
That makes Ax = b solvable, so b is in the column space. All columns of A pass this
test b3 +b2 −5b1 = 0. This is the equation for the plane (in the first description of
the column space).
My question is this: is correct to say that “The column space of A contains all vectors with b3+b2−5b1 = 0”?
I know that all the vectors in column space of A will pass this test b3+b2-5b1 but wont there be vectors outside the column space of A which also pass this test? So, should we say ” the column space of A contains all vectors which pass the test“?