Linear Algebra, subspaces and row reducing

[flyingpig]

Homework Statement

This is just a conceptual question

Whenever you are asked for a basis for the subspace spanner by some set of vectors, is that the same as asking the basis that forms the column space of that matrix? Are the dimension for that subspace the same as the column space?

The other thing bothers me is that why is it that we have to put these vectors in a matrix and row reduce it to see the pivot columns, what is the magic in row reducing that tells us which vector is "redundant" in the set?

 

[lineintegral1]

If the basis of your subspace is contained within a matrix, then yes, it is the basis of the column space. In other words, it is the image of the matrix. And yes, the dimension of the image is the dimension of the subspace.

For your last question, have you looked to the Rank-Nullity Theorem? This should give you an answer to your question. Think about how the pivots of a matrix relate to the rank.