Good times with Linear Algebra

Identity

Identity matrix of size 1:
Identity matrix of size 2:
Identity matrix of size 3:
Identity matrix of size 4:
Identity matrix of size 5:
Cross product of [1] and itself:
Cross product of [1,0, 0,1] and itself:
Cross product of [1,0,0, 0,1,0, 0,0,1] and itself:

More Linear Algebra

Cross product of [2,3, -2,-1] and [4,0, 1,2]:
Cross product of [1,2,3, 1,2,3, 1,2,3] and [0,1,2, -1,-1,-1, 1,1,1]:

Rotations

Remember that sin(π/4)=√(½)=

Rotations around the x-axis

Vector [0,1,0] rotated 0:
Vector [0,1,0] rotated π/4:
Vector [0,1,0] rotated π/2:
Vector [0,1,0] rotated π:
Vector [0,1,0] rotated -π:
Vector [0,1,0] rotated 3π/2:
Vector [0,1,0] rotated 2π:
Vector [0,1,0] rotated π/4 and again by π/4:
Vector [0,1,0] rotated π/4 and again by -π/4:

Rotations around the y-axis

Vector [1,0,0] rotated 0:
Vector [1,0,0] rotated π/4:
Vector [1,0,0] rotated π/2:
Vector [1,0,0] rotated π:
Vector [1,0,0] rotated -π:
Vector [1,0,0] rotated 3π/2:
Vector [1,0,0] rotated 2π:
Vector [1,0,0] rotated π/4 and again by π/4:
Vector [1,0,0] rotated π/4 and again by -π/4:

Rotations around the z-axis

Vector [1,0,0] rotated 0:
Vector [1,0,0] rotated π/4:
Vector [1,0,0] rotated π/2:
Vector [1,0,0] rotated π:
Vector [1,0,0] rotated -π:
Vector [1,0,0] rotated 3π/2:
Vector [1,0,0] rotated 2π:
Vector [1,0,0] rotated π/4 and again by π/4:
Vector [1,0,0] rotated π/4 and again by -π/4:

Scaling

Vector [1,1,1] scaled 2× in x direction:
Vector [1,1,1] mirrored -2× in y direction:
Vector [1,1,1] scaled ½× in z direction:
Vector [1,1,1] scaled 1× in each direction:
Vector [1,1,1] mirrored -1× in each direction:
Vector [1,1,1] scaled 0× in each direction:
Vector [1,1,1] scaled 1000× in each direction:
Vector [1,1,1] scaled by 2× then again ½× in each direction:
Vector [1,1,1] scaled by 2× then again 3× in each direction:

Translation

Vector [0,0,0] translated by 8 along x-axis:
Vector [0,0,0] translated by -8 along y-axis:
Vector [0,0,0] translated by 8 along z-axis:
Vector [0,0,0] translated by 0:
Vector [0,0,0] translated by 8 along each axis:
Vector [0,0,0] translated by 8 and then again by -8 along each axis:
Vector [0,0,0] translated by 4 and then again by 4 along each axis:
Vector [0,0,0] translated by 8 in x and then again by 5 in y:

Multiple transformations applied to a vector

(RR) Vector [1,0,0] rotated π/2 around z-axis (result is ), then again π/2 around the x-axis.
(RS) Vector [1,1,1] rotated π/2 around z-axis (result is ), then scaled 2× along x direction.
(SR) Vector [1,1,1] scaled 2× along x direction (result is ), then rotated π/2 around z-axis.
(ST) Vector [1,1,1] scaled 2× in each direction (result is ), then translated by 1 in y direction:
(TS) Vector [1,1,1] translated by 1 in y direction (result is ), then scaled 2× in each direction:
(TRT) Vector [0,0,0] translated 1 in y, then rotated π around z, then translated 1 in y again:
(TRTR) Vector [0,1,0] translated 1 in y, rotated π/2 around z-axis, translated 1 in y again, and finally rotated π/2 around z again:
(RTRT) Vector [0,1,0] rotated π/2 around z-axis, translated 1 in y, rotated π/2 around z again, and finally translated 1 in y again:

Concatenated Matrices

(RT) Build a transformation matrix that first rotates π/2 around z then translates 1 in y. Apply to [0,0,0]:
(TT) Build a transformation matrix that first scales 2× in each direction then translates 1 in y. Apply to [0,0,0]:

More information

See also Concatenating Transforms for more information about how to perform local and global transformations.