C has the same number of rows
as A and the same number of columns
as B, and
each column j of C is from
column j of B acting as
coefficients in a linear
combination of all the columns
of A.
Next: For the set of
real numbers R,
positive integers m and n,
m by n real matrix A,
real numbers a and b,
and n by 1 x and y,
we have that
A(ax + by) = (aA)x + (bA)y
so that A is linear.
If we regard x and y as vectors
in the
n-dimensional real vector space
R^n, then we have that
A is a function
A: R^n --> R^m
and a linear operator
which can be good to know.
Then the subset of R^n
K = { x | x in R^n and Ax = 0 }
(where 0 is the m by 1 matrix of all
zeros)
is important to understand.
E.g., K = [0] if and only if
function A is 1-1. If m = n,
then A has an inverse
A^(-1) if and only if A is 1-1.
With C = AB, the matrix
product is the same as
function composition so that
Cx = (AB)x = A(Bx)
which also can be good to know and,
of course, uses just the
associative law of matrix
multiplication.
That matrix multiplication is
associative is a biggie --
sometimes says some big things,
e.g., is the core of duality
theory in linear programming.
And the situation is entirely
similar for the complex numbers
C in place of the real numbers R.
AB = C,
C has the same number of rows as A and the same number of columns as B, and each column j of C is from column j of B acting as coefficients in a linear combination of all the columns of A.
Next: For the set of real numbers R, positive integers m and n, m by n real matrix A, real numbers a and b, and n by 1 x and y, we have that
A(ax + by) = (aA)x + (bA)y
so that A is linear.
If we regard x and y as vectors in the n-dimensional real vector space R^n, then we have that A is a function
A: R^n --> R^m
and a linear operator which can be good to know.
Then the subset of R^n
K = { x | x in R^n and Ax = 0 }
(where 0 is the m by 1 matrix of all zeros) is important to understand. E.g., K = [0] if and only if function A is 1-1. If m = n, then A has an inverse A^(-1) if and only if A is 1-1.
With C = AB, the matrix product is the same as function composition so that
Cx = (AB)x = A(Bx)
which also can be good to know and, of course, uses just the associative law of matrix multiplication.
That matrix multiplication is associative is a biggie -- sometimes says some big things, e.g., is the core of duality theory in linear programming.
And the situation is entirely similar for the complex numbers C in place of the real numbers R.