The document discusses matrices and linear algebra concepts such as: - A matrix is a set of elements organized into rows and columns. Basic matrix operations include addition, subtraction, and multiplication. - Vectors can be represented as matrices and operations like the dot product and cross product are used to describe relationships between vectors. - Important matrix properties include inverses, determinants, and homogeneous matrices which allow translations and perspective transforms. - An orthonormal basis is a set of orthogonal vectors that form a coordinate system where the magnitude of each basis vector is 1.

1. Understand of linear algebra
Edward J. Yoon (edward@udanax.org)
April 18, 2008

2. What is the Matrix?
• A matrix is a set of elements, organized
into rows and columns
rows
a b 
c d 
columns
 

3. Matrix Operations
f  a  e b  f 
a b   e
c d    g   c  g d  h  Just add elements
  h  
f  a  e b  f 
a b   e
c d    g   c  g d  h  Just subtract elements
  h  
f  ae  bg af  bh
a b   e Multiply each row
  ce  dg
c d   g cf  dh  by each column
  h  

4. Vector Operations
• Vector: 1 x N matrix
a 
• Interpretation: a line
 
in N dimensional
v  b 
space
• Dot Product, Cross
c 

Product, and
Magnitude defined
y
on vectors only
v
x

5. Vector Interpretation
• Think of a vector as a line in 2D or 3D
• Think of a matrix as a transformation on
a line or set of lines
 x  a b   x'
 y   c d    y '
   

6. Dot Product
• Interpretation: the dot product measures
to what degree two vectors are aligned
A
A+B = C
(use the head-to-tail
B method to combine vectors)
C
B
A

7. Dot Product
d 
a  b  abT  a b c  e   ad  be  cf Think of the dot product

as a matrix multiplication
f

The magnitude is the dot
a  aaT  aa  bb  cc
2
product of a vector with itself
The dot product is also related to
a  b  a b cos( )
the angle between the two
vectors – but it doesn’t tell us the
angle

8. Cross Product
• The cross product of vectors A and B is a vector C
which is perpendicular to A and B
• The magnitude of C is proportional to the cosine of
the angle between A and B
• The direction of C follows the right hand rule – this
why we call it a “right-handed coordinate system”
a  b  a b sin( )

9. Inverse of a Matrix
• Identity matrix:
AI = A
• Some matrices have
1 0 0
an inverse, such that:
0 1 0 
AA-1 = I
I 
• Inversion is tricky:

(ABC)-1 = C-1B-1A-1
0 0 1 
 
Derived from non-
commutativity
property

10. Inverse of a Matrix
1. Append the identity
matrix to A
2. Subtract multiples of the
a b c 1 0 0
other rows from the first
d e 
f  0 1 0 row to reduce the
 diagonal element to 1
g h i 0 0 1
  3. Transform the identity
matrix as you go
4. When the original matrix
is the identity, the
identity has become the
inverse!

11. Determinant of a Matrix
• Used for inversion a b 
A
• If det(A) = 0, then A c d
 
has no inverse
• Can be found using det( A)  ad  bc
factorials, pivots, and
cofactors! 1  d  b
1
A
• Lots of ad  bc  c a 
 
interpretations – for
more info, take 18.06

12. Determinant of a Matrix
ab c
f  aei  bfg  cdh  afh  bdi  ceg
de
gh i
ab ca b ca b c
Sum from left to right
de fd e fd e f Subtract from right to left
Note: N! terms
gh ig h ig h i

13. Homogeneous Matrices
• Problem: how to include translations in
transformations (and do perspective
transforms)
• Solution: add an extra dimension
1  x 
1  
y
x z 1  
y
 1  z
  
 1  1

14. Ortho-normal Basis
• Basis: a space is totally defined by a set
of vectors – any point is a linear
combination of the basis
• Ortho-Normal: orthogonal + normal
• Orthogonal: dot product is zero
• Normal: magnitude is one
• Example: X, Y, Z (but don’t have to be!)

15. Ortho-normal Basis
x  1 0 0 x y  0
T
y  0 1 0 xz  0
T
z  0 0 1 yz  0
T
X, Y, Z is an orthonormal basis. We can describe any
3D point as a linear combination of these vectors.
How do we express any point as a combination of a
new basis U, V, N, given X, Y, Z?

16. Ortho-normal Basis
n1  a  u  b  u  c  u 
a 0 0  u1 v1
0 b 0 u n2    a  v  b  v  c  v 
v2
  2  
 0 0 c   u3 n3  a  n  b  n  c  n
   
v3
(not an actual formula – just a way of thinking about it)
To change a point from one coordinate system to
another, compute the dot product of each
coordinate row with each of the basis vectors.