1. Math review - scalars, vectors, and matrices
Matt Boggus
CSE 3541/5541

2. Outline Coordinate Spaces Primitives Operations Matrices Scalars
Vectors
Points
Operations
Matrices

3. Boardwork examples: 1D, 2D, 3D coordinate systems
Instances of primitives are defined within the context of a coordinate system, so refresh with examples of these first.

4. Coordinate system in Unity
3D, left-handed coordinate system

5. Scalars Scalar value – a number (integer or real) ; one unit of data
Ex: the scale, weight, or magnitude of something
Examples elsewhere in CS&E
(Software I) 2 scalars -> low and high for random number range
(Systems I) Representing fractional values using binary numbers
Au’16 deleted slides on qualities and properties of scalar addition and multiplication

6. Vectors Vector – a quantity possessing both magnitude and direction
Usually visualized as a directed line segment, beginning at the origin, but technically have no start position
Mathematically described as n-tuples of scalars
Examples: (0,0,0) (x,y) (r,g,b,a)

7. Vector operations – mathematically
Vectors = n-tuples
Vector-vector addition
Scalar-vector multiplication
Vector decomposition
Numeric examples of vector operations

8. Vector operations – intuitively
Vector-Vector Addition
Visualize using head-to-tail axiom
Scalar-vector multiplication
Resize (scale) the
directed line segment
α > 1 length increases
0 > α > 1 length decreases
α < 0 reverse direction
Head-to-tail axiom
Scalar-vector multi.

9. Points Point – a location or position
Visualized as a dot relative to an origin
Mathematically described as n-tuples of scalars
Examples: (0,0,0) (x,y) (r,g,b,a)
Note: points and vectors have the same representation!

10. Coordinate system or frame
Origin – a point
Indicates the “center” of the coordinate systems
Other points are defined relative to the origin
Basis vectors
A set of linearly independent vectors
None of them can be written as a linear combination of the others
Basis
vectors
located
at the
origin
Arbitrary
placement
of basis
vectors

11. Vectors and points in a coordinate system
Coordinate frame defined by point P0 and set of vectors
A vector v is
A point P is
Note: α and β indicate scalar values

12. Practice problem
Given a point P = (2,-5,1) in a right handed coordinate system, what is the point is a left handed coordinate system?

13. Point-Vector operations
P and Q are points, v is a vector
Point-point subtraction operation
Vector-point addition operation

14. Working toward a distance metric
The previous operations do not include a way to estimate distance between points
Create a new operation: Inner (dot) Product
Input: two vectors Output: scalar
Properties we want:
For orthogonal vectors

15. Dot product logic If we can multiply two n-tuples, this implies
Magnitude (length) of a vector
Distance between two points
Note: some reference materials use || instead of |

16. Dot product computation
Dot product of two vectors, u and v
Ex: (5,4,2) ∙ (1,0,1) = 5 * * * 1 = 7
With some algebra we find that the dot product is equivalent to where θ is the angle between the two vectors
cosθ = 0  orthogonal
cosθ = 1  parallel

17. Projections
We can determine if two points are “close” to each other, what about vectors?
How much of w is in the same direction as v?
Given vectors v and w, decompose w into two parts, one parallel to v and one orthogonal to v
Step 1: decompose w into two parts, one with some unknown amount (alpha) in the same direction as v (hence the scaling of v by alpha) plus some vector u whose direction we define as orthogonal to v but has an unknown magnitude.
Step 2: Dot product w and v, but substitute the decomposition of w and distribute.
Step 3 (same line as step 2): by definition u dot v is zero and drops out
Step 4: solve for alpha
Step 5: solve for u
Projection of one
vector onto another

18. Cross product Input: two vectors
Output: vector, orthogonal to input vectors
General formula
Two 3D vectors
Note: sin and cos (also sin-1 and cos-1)can be expensive to compute – involves computing a series until sufficient precision is reached
See After computers: power series on page
Modified Figure from Essential Mathematics for Games and Interactive Applications

19. Matrices Definitions Matrix Operations Row and Column Matrices
Change of Representation
Relating matrices and vectors

20. What is a Matrix?
A matrix is a set of elements, organized into rows and columns
rows
columns

21. Definitions n x m Array of Scalars (n Rows and m Columns)
n: row dimension of a matrix, m: column dimension
m = n  square matrix of dimension n
Element
Transpose: interchanging the rows and columns of a matrix
Column Matrices and Row Matrices
Column matrix (n x 1 matrix):
Row matrix (1 x n matrix):

22. Matrix Operations Scalar-Matrix Multiplication Matrix-Matrix Addition
Multiply every element by the scalar
Matrix-Matrix Addition
Add elements with same index
Matrix-Matrix Multiplication
A: n x l matrix, B: l x m  C: n x m matrix
Easy to overlook that cij is a single element, so computing C requires iterating over rows and columns.
cij = the sum of multiplying elements in row i of matrix a times elements in column j of matrix b

23. Matrix Operation Examples

24. Matrix Operations Properties of Scalar-Matrix Multiplication
Properties of Matrix-Matrix Addition
Commutative:
Associative:
Properties of Matrix-Matrix Multiplication
Identity Matrix I (Square Matrix)

25. Matrix Multiplication Order
Is AB = BA? Try it!
Matrix multiplication is NOT commutative!
The order of series of matrix multiplications is important!
Generalize where possible, but don’t forget about special cases

26. Inverse of a Matrix Identity matrix: AI = A
Some matrices have an inverse, such that: AA-1 = I

27. Inverse of a Matrix
Do all matrices have a multiplicative inverse? Consider this example, try to solve for A-1:
AA-100 =
Given A, try to solve for the 9 variables in its inverse A-1 by setting the result of the multiplication AA-1 equal to I. If no solution exists, the matrix has no inverse.
0 * a + 0 * d + 0 * g = 0 ≠ 1
Note: 1 is the element at 00 in the identity matrix

28. Inverse of Matrix Concatenation
Inversion of concatenations
(ABC)-1 = ?
A * B * C * X = I
A * B * C * C-1 = A * B
A * B * B-1 = A
A * A-1 = I
Order is important, so X = C-1B-1A-1
Given A, B, and C assuming each has an inverse, we can construct the inverse of a series of multiplications ABC by constructing a matrix such that each individual “cancels” with its inverse.

29. Row and Column Matrices + points
By convention we will use column matrices for points
Column Matrix
Row matrix
Concatenations
Associative
By Row Matrix

30. Summary Primitives: scalars, vectors, points
Operations: addition and multiplication
Matrix representation and operations