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

Outline Coordinate Spaces Primitives Operations Matrices Scalars Vectors Points Operations Matrices

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.

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

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

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)

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

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.

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!

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

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

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?

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

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

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 |

Dot product computation Dot product of two vectors, u and v Ex: (5,4,2) ∙ (1,0,1) = 5 * 1 + 4 * 0 + 2 * 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

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

Cross product Input: two vectors Output: vector, orthogonal to input vectors General formula Two 3D vectors http://en.wikipedia.org/wiki/Cross_product 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 http://www2.clarku.edu/~djoyce/trig/compute.html Modified Figure 1.1.6 from Essential Mathematics for Games and Interactive Applications

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

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

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):

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

Matrix Operation Examples

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

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

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

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

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.

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

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