Question

In lecture 1, Prof. Strang asks 'Do the linear combination of columns fill the 3D space?'. Is he asking if 'b' has non-zero components on x, y, and z axes?

Answer 1

I'm assuming he's referring to a 3 by 3 matrix. If so, then he's asking if you could construct any vector in 3D space from the column vectors of the matrix. So, given any vector 'b' on the right hand side of the equation Ax=b, can you always find a solution 'x' that satisfies the equation. If so, the linear combination of columns fill the space, otherwise they don't. You'll see later that this is also equivalent to asking whether or not the three vectors (in this case of a square 3 by 3 matrix) are linearly independent.

I hope this helped.

Answer 2

vsharma, ovanbc's answer is exactly correct as to what Prof. Strang is asking. Note that he is "jumping ahead" several lectures at that point, so if you don't fully understand his question, don't worry, he will explain it more fully when you get into lectures 6, 7, and 8 about null spaces and column spaces and such.

But, to answer your second question specifically: no that's not what he's trying to say, in fact just the opposite. Notice that the 3-dimensional zero vector [0 0 0], if you plotted it onto a graph, would have 0 components on all three axes, but the zero vector is itself part of 3-d space. In fact, it's very important that columns of the matrix A *do* have at least one combination that produces 0, because that's how we find the "null space" (where Ax = 0).

The key here is that "all of three-d space" means every possible three-dimensional point, and that includes the point (0,0,0).