But youhaven'tincluded the condition at all. A general vector of that form would be (2A, B, 2C) and those donotsatisfy (2A)+4(B)-3(2C)= 0. In order that (A, B, C) satisfy A+ 4B- 3C= 0 then you must have A= 3C- 4B. In particular, if C= 1 and B= 0, A= 3. <3, 0, 1> satisfies (3)+ 4(0)-3(1)= 0. Also if C= 0 and B= 1, A= -4. <-4, 1, 0> satisfies (-4)+ 4(1)- 3(0)= 0.

The dimension is NOT 3, it is 2. And I know that because I just wrote a basis.

Once again, such a vector does NOT satisfy the condition you gave.Once again .

However, i'm a little puzzled regarding this basis. Would it be:

?