**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:

begin{bmatrix}1&2&3&5&b1\2&4&8&12&b2\3&6&7&13&b3end{bmatrix}

After applying gaussian elimination:

begin{bmatrix}1&2&3&5&b1\0&0&2&2&b2-2b1\0&0&0&0&b3+b2-5b1end{bmatrix}

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**“?