A line in R^{n} always may be represented as the set of points of the form a + t v where a is any point on the line, v = b - a, with b any other point on the line, and the variable t, called a parameter, varies over the set of all real numbers.
A plane in R^{n} (for n >= 3) always may be represented as the set of points of the form a + s v + t w where a is any point in the plane, v = b - a, w = c - a, with b and c any points in the plane such that the three points a, b, c do not lie on a common line, and the variables s and t, called parameters, vary over the set of all real numbers.
Find a parametric representation of the plane in space consisting of all points (x, y, z) such that x + 3 y - 2 z = 2.
Use row operations on the multiply-augmented matrix (A 1_{2}), where 1_{2} is the 2 \times 2 identity matrix, to attempt inversion of the following matrix A:
| a | b |
| c | d |
Let f(x) = M x where M is the 4 \times 4 matrix
| 1 | 1 | -2 | -1 |
| 3 | -5 | -4 | 3 |
| 2 | -2 | -3 | 1 |
| 4 | -8 | -5 | 5 |
Determine if M is an invertible matrix, and, if it is, find its inverse.
What is the kernel of f ?
What is the image of f ?
Determine the fiber of f over the point (1, 1, 1, 1).