18.06 Linear Algebra, Spring 2010
Transcript – Lecture 13
OK. Uh this is the review lecture for the first part of the course, the Ax=b part of the
course. And the exam will emphasize chapter three. Because those are the --0
chapter three was about the rectangular matrices where we had null spaces and null
spaces of A transpose, and ranks, and all the things that are so clear when the
matrix is square and invertible, they became things to think about for rectangular
matrices. So, and vector spaces and subspaces and above all those four subspaces.
OK, I'm thinking to start at least I'll just look at old exams, read out questions, write
on the board what I need to and we can see what the answers are.
The first one I see is one I can just read out.
Well, I'll write a little. Suppose u, v and w are nonzero vectors in R^7. What are the
possible -- they span a -- a vector space. They span a subspace of R^7, and what
are the possible dimensions? So that's a straightforward question, what are the
possible dimensions of the subspace spanned by u, v and w? OK, one, two, or three,
right. One, two or three.
Couldn't be more because we've only got three vectors, and couldn't be zero because
-- because I told you the vectors were nonzero. Otherwise if I allowed the possibility
that those were all the zero vector -- then the zero-dimensional subspace would
have been in there. OK.
Now can I jump to a more serious question? OK. We have a five by three matrix.
And I'm calling it U. I'm saying it's in echelon form. And it has three pivots, r=3.
OK. First question, what's the null space? What's the null space of this matrix U, so
this matrix is five by three, and I find it helpful to just see visually what five by three
means, what that shape is. Three columns.
Three columns in U then, five rows, three pivots, and what's the null space? The null
space of U is -- and it asks for a spec-of course I'm looking for an answer that isn't
just the definition of the null space, but is the null space of this matrix, with this
information. And what is it? It's only the zero vector.
Because we're told that the rank is three, so those three columns must be
independent, no combination -- of those columns is the zero vector except -- so the
only thing in this null space is the zero vector, and I -- I could even say what that
vector is, zero, zero, zero. That's OK.
So that's what's in the null space.
All right? let me go on with -- this question has multiple parts. What's the -- oh now
it asks you about a ten by three matrix, B, which is the matrix U and two U. It