Geometry of linear equations: matrix as a mapping
General idea. Let be a function with the domain We would like to know for which the equation has solutions, and if it does, then how many (one or more). For the existence part, let the argument run over and see what set the values belong to. It is the image This definition directly implies
Basic observation 1. The equation has solutions if and only if
For the uniqueness part, fix and see for which arguments the value equals that This is the counter-image of
Basic observation 2. If consists of one point, you have uniqueness.
See how this works for the function This function is not linear. The function generated by a matrix is linear, and a lot more can be said in addition to the above observations.
Matrix as a mapping
Definition 1. Let be a matrix of size It generates a mapping according to ( is written as a column). Following the common practice, we identify the mapping with the matrix
Exercise 1 (first characterization of matrix image) Show that the image consists of linear combinations of the columns of
Solution. Partitioning into columns, for any we have
This means that when runs over the images are linear combinations of the column-vectors
Exercise 2. The mapping from Definition 1 is linear: for any vectors and numbers one has
Proof. By (1)
Remark. In (1) we silently used the multiplication rule for partitioned matrices. Here is the statement of the rule in a simple situation. Let be two matrices compatible for multiplication. Let us partition them into smaller matrices
Then the product can be found as if those blocks were numbers:
The only requirement for this to be true is that the blocks be compatible for multiplication. You will not be bored with the proof. In (1) the multiplication is performed as if were numbers.