Download 20 The Column Space

The Column Space of a Matrix
Learning Goal: students see one of the important subspaces tied to a matrix.
One very important example of a subspace comes from a matrix A. We started by talking
about linear combinations of the columns of a matrix. If A is m × n, then the each column is in
Rm (assuming real stuff for now). Consider the set of all linear combinations of the columns of
the matrix A. This is a subspace of Rm. It is called the column space of A, denotes C(A). (Other
notations exist, so be careful!)
Why is it a subspace? Well, it is non-empty. If the columns are c1, c2, …, cn then we can
certainly take 0c1 + 0c2 + ! + 0cn = 0. And it is closed under addition and scalar multiplication:
(a1c1 + ! + ancn) + (b1c1 + ! + bncn) = (a1 + b1)c1 + ! + (an + bn)cn and k(a1c1 + ! + ancn) is
easily seen to be (ka1c1 + ! + kancn), all of which are linear combinations of the columns of the
matrix.
Another way to think of the column space is that we take Ax for all possible x’s. Thus,
the column space is like taking the image of the function f(x) = Ax (in fact, some texts use the
notation ℜ(A) to denote the “range”). In this way, solving Ax = b is like testing whether b is in
the image. A system Ax = b is solvable if and only if b is in the column space of A.
⎡ 1 0 3⎤
Example: find the column space of ⎢
⎥ . In other words, find all possible linear
⎣2 1 1⎦
⎡1 ⎤
⎡0 ⎤
⎡ 3⎤
combinations a ⎢ ⎥ + b ⎢ ⎥ + c ⎢ ⎥ . Well, we can set a to be anything we want, and get all
⎣2 ⎦
⎣1 ⎦
⎣1 ⎦
⎡a⎤
vectors of the form ⎢ ⎥ . But then by judicious choice of b we can change the second
⎣ 2a ⎦
component to be anything without affecting the first component. In some sense c is just a red
herring, something that gives us extra latitude in solving systems involving this matrix of
coefficients (note that there will be more variables than equations, so it is likely that there will be
many solutions for every right-hand side). So we have determined that C(A) = R2. Note this is
equivalent to being able to solve Ax = b for any right-hand side b.
⎡1 2 ⎤
Example: find the column space of the transpose of this matrix, ⎢⎢ 0 1 ⎥⎥ . Now there are more
⎢⎣ 3 1 ⎥⎦
equations than variables, so it is likely that we won’t be able to solve for all right-hand sides.
Naturally we can solve for some (like 0, or (1, 0, 3)). In fact, the column space is the set of
linear combinations of two vectors in R3, which we expect to be a two-dimensional set—a plane.
In fact, the column space is the plane 3x – 5y – z = 0. Notice that this plane goes through the
origin, as it must to have any chance of being a subspace.
Read: 3.1
Problems: 3.1: 1, 2, 4, 5, 7, 8, 10, 11, 12, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32