Definition: A subset Pi of R^{n} is called a linear subspace of R^{n} if (1) it contains p + q whenever it contains p and q and (2) if it contains every scalar multiple s p of any of its points p.
Theorem: A subset Pi of R^{n} is a linear subspace of R^{n} if and only if there is a number k and an n \times k matrix C such that Pi is the image of the map gamma from R^{k} to R^{n} given by gamma(t) = C t for t in R^{k}.
The map gamma amounts to the same thing as a parameterization of Pi in which the coordinates of t are the parameters. If the columns of the matrix C are C_{1}, ..., C_{k}, then the point x = C t of Pi has the form
Such an expression x is called a linear combination of the columns of C.
With the previous notation the image of gamma (a subspace of R^{n}) is called the column space of the matrix C. It is the set of all linear combinations of the columns of C.
Definition: For a given linear subspace Pi of R^{n} the smallest possible number of parameters (in any parameterization as above) is called the dimension of Pi.
A. Let M be the 3 \times 4 matrix that is given by
| -2 | 1 | -3 | 2 |
| 1 | 0 | 1 | -1 |
| 3 | -1 | -3 | 4 |
) .
What is the column space of M ?
Find a 3 \times 3 matrix that has the same column space as M.
B. Let N be the 3 \times 4 matrix that is given by
| -2 | 1 | -3 | -1 |
| 1 | 0 | 1 | 1 |
| 3 | -1 | 4 | 2 |
) .
What is the column space of N ?
What is the smallest possible size of a matrix having the same column space as N ?
What is the dimension of the column space of N ?
Does N have the same column space as its reduced row echelon form?