f is a linear-function (-map, -transformation, ...) iff

f(v + w) = f(v) + f(w),

f(k v) = k f(v)

(consequently

f(0) = 0.)

f is often called a linear-transformation if the input and output spaces of f are the same.

A linear transformation over a finite-dimension vector-space can be represented by a matrix.

Examples

Identity, I:

I

=

1 0 0 1

I

x y

=

x y

 

Projection of <x, y>, onto the line through the origin whose unit normal is n = <n_x, n_y>:

1-nx2 -nxny -nxny 1-ny2

x y

(<x, y> → <x, y> - (<x, y> . n) n = <x, y> - (x n_x - y n_y) n = <x - (x n_x + y n_y) n_x, y - (x n_x + y n_y) n_y>.)

 

Reflection of <x, y>, in the line through the origin whose unit normal is n = <n_x, n_y>:

1-2nx2 -2nxny -2nxny 1-2ny2

x y

 

Anti-clockwise rotation of <x, y>, by angle θ, about the origin:

cosθ - sinθ sinθ cosθ

x y

Equations

Given constants a, b, c, d, p, q, s, and t, and variables w, x, y, and z, in matrices

a b c d

w x y z

=

p q r s

i.e.,

a w + b y = p,

a x + b z = q,

c w + d y = r,

c x + d z = s.

 

The following are all 2×2 but generalize ...

 

a b c d w x y z = p q r s	is equivalent (wrt solving for w, x, y, and z) to	b a d c y z w x = p q r s col swap row swap
to
c d a b w x y z = r s p q row swap   row swap
to	and to
ka kb c d w x y z = kp kq r s row multiplication by constant k	a+kc b+kd c d w x y z = p+kr q+ks r s row addition (or subtraction)

 

If we can work on M and P to reduce

M X = P

to an equivalent

I X = P',

using the relations above, then we can just read off the solution, P', for X.

X and P can be n×1 column vectors, or n×n matrices, etc..

For example, let P=I, the identity, and reduce

M X = I

to

I X = M^-1

giving the matrix inverse of M.

(Note that any column swaps cause row swaps, in X, which must be undone to get the final answer.)