A plane Pi in R^{n} that passes through the origin has two arithmetic properties that distinguish it from a plane that does not pass through the origin:
If P and Q are points of Pi, then P + Q is also a point of Pi.
If c is a number and P is a point of Pi, then the scalar multiple c P of P is also a point of Pi.
Any subset Pi of the Cartesian space R^{n} that satisfies the two rules just stated (in the special case of a plane through the origin) is called a linear subspace of R^{n}.
Re-statement of the first item above: A plane in R^{n} is a linear subspace if and only if it passes through the origin.
A line in R^{n} is a linear subspace if and only it passes through the origin.
If M is an m \times n matrix and f is the function from R^{n} to R^{m} that is given by f(x) = M x for x in R^{n}, then the image of f is a linear subspace of R^{m}.
Find two perpendicular vectors of the same length in the plane that is the linear subspace of R^{3} consisting of the points (x, y, z) for which
Let f(x) = M x where M is the 4 \times 4 matrix
| 1 | 1 | -2 | 0 |
| 2 | -2 | -3 | 1 |
| 3 | -5 | -4 | 2 |
| 1 | -3 | -1 | 1 |
Explain why the kernel of f must be a linear subspace of R^{4}. What is its dimension?