If
x = (x_{1}, x_{2}, ..., x_{M}) and y = (y_{1}, y_{2}, ..., y_{M})
are two vectors with the same number of coordinates, then their inner product (also called scalar product or dot product) is the numberx . y = x_{1} y_{1} + x_{2} y_{2} + ... + x_{M} y_{M} .
If A is an L \times M matrix and B is an M \times N matrix, then there is a product matrix AB that is an L \times N matrix. The number in row i and column j of the product AB is defined to be the inner product of the i^{th} row of A with the j^{th} column of B.
A system of M linear equations in N variables may be regarded as a single matrix equation
A x = b
where A is M \times N, x is a column of N variables and b is a column of M numbers.If A is an M \times N matrix and x is a column of N variables, then the formula
f(x) = A x
defines a function f.This function f satisfies the rules:
f(x + x') = f(x) + f(x') for any vectors x and x'.
f(a x) = a f(x) for any number a and any vector x.
The exercises below pertain to the function:
(u, v, w) = f(x, y, z)
where
| u = x - 2 y - z |
| v = 5 x + 4 y - 3 z |
| w = - 2 x - 3 y + z |
Find all points (x, y, z) for which f(x, y, z) = (1, -5, 3).
Describe the set of all points (x, y, z) for which f(x, y, z) = (1, -5, 3) as a subset of space.
Under what condition on (u, v, w) is it the case that (u, v, w) is the value of f at some point?
Describe the set of all values of f as a subset of space.