Beginning Direct3D Game Programming:Mathematics 5

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Division of Digital Contents, DongSeo University.15, May 2016

Linear system

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A linear function is a polynomial function of degree zero or one, or is the zero polynomial(f(x)=0).

Linear function When the function is of only one variable, it is of the

form

where a and b are constants, often real numbers. The graph of such a function of one variable is a nonver-

tical line. a is frequently referred to as the slope of the line, and b as the intercept.

For a function  of any finite number of independent variables, the general formula is

and the graph is a hyperplane of dimension k.

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Linear map In linear algebra, a linear function is a map f between

two vector spaces that preserves vector addition and scalar multiplication:

This tells us that for a some input, if we can decompose the input to addition, the result also be calculated sepa-rately.

In physics and other sciences, a nonlinear system, in contrast to a linear system, is a system which does not satisfy the above properties – meaning that the output of a nonlinear system is not directly proportional to the input.4

Linear system A system of linear equations (or linear system) is a

collection of two or more linear equations involving the same set of variables.

Above is a system of three equations in the three vari-ables x, y, z.

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Plane A plane is a flat, two-dimensional surface that extends

infinitely far.

The equation of a plane with nonzero normal vector n=(a,b,c) through the point x0=(x0,y0,z0) is n·(x-x0)=0, where x=(x,y,z).

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Plugging in gives the general equation of a plane,ax+by+cz+d=0,

whered=-ax0-by0-cz0.

A plane specified in this form therefore has x-, y-, and z-intercepts atx = -d/ay = -d/bz = -d/c,

and lies at a distanceD=d/(sqrt(a2+b2+c2))

from the origin.

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Plane Point Distance

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Linear system

A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied.

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A linear system in three vari-ables determines a collection of planes. The intersection point is the solution.

Matrix

Each element of a matrix is often denoted by a variable with two subscripts. For instance, a2,1 represents the el-ement at the second row and first column of a matrix A.

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A matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The dimensions of below matrix are 2 × 3 (read "two by three"), because there are two rows and three columns.

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Size The size of a matrix is defined by the number of rows

and columns that it contains. A matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, while m and n are called its dimensions.

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Notation Matrices are commonly written in box brackets or 

parentheses:

The (1,3) entry of the following matrix A is 5. It is de-noted a13, a1,3, A[1,3] or A1,3.

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Basic operations There are a number of basic operations that can be ap-

plied to modify matrices.– matrix addition– scalar multiplication– transposition– matrix multiplication– row operations.

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Addition The sum A+B of two m-by-n matrices A and Bis calcu-

lated entrywise: (A + B)i,j = Ai,j+ Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.

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Transposition The transpose of an m-by-n matrix A is the n-by-m ma-

trix AT formed by turning rows into columns and vice versa:

(AT)i,j = Aj,i

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Scalar multiplication The product cA of a number c (also called a scalar in the

parlance of abstract algebra) and a matrix A is com-puted by multiplying every entry of A by c:

(cA)i,j = c · Ai,j. This operation is called scalar multiplication.

– Its result is not named “scalar product” to avoid confusion, since “scalar product” is sometimes used as a synonym for “inner product”.

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Multiplication Multiplication of two matrices is defined if and only if the

number of columns of the left matrix is the same as the number of rows of the right matrix.

If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose en-tries are given by dot product of the corresponding row of A and the corresponding column of B:

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Schematic depiction of the matrix product AB of two matrices A and B.

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example

Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A+B)C = AC+BC as well as C(A+B) =CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined. The product AB, they need not be equal, that is, generally

AB ≠ BA,21

3=2×0+3×1+4×0 2340=2×1000+3×100+4×10

Identity element, Inverse element for multiplica-tion For some real number a a×1 = a In this case, 1 is called the identity element for multi-

plication.

a×( ) = 1 The answer to above equation is 1/a, 1/a is called the

inverse element of a for multiplication.

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Identity Matrix In linear algebra, the identity matrix, or sometimes

ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

When A is m×n, it is a property of matrix multiplication that

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Determinant In linear algebra, the determinant is a useful value that

can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A) or |A|.

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Linear Systems Matrices provide a compact and convenient way to rep-

resent systems of linear equations. For instance, the lin-ear system is given below.

Above linear system can be represented in matrix form like below.

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The matrix preceding the vector x, y, z of unknowns is called the coefficient matrix, and the column vector on the right side of the equals sign is called the con-stant vector. Linear systems for which the constant vector is nonzero (like the example above) are called non-homogeneous.

Linear systems for which every entry of the constant vector is zero are called homogeneous.– The geometric meaning of homogeneous is all the 3-planes

meet at the origin (0,0,0).26

How to solve? We can set the augmented matrix formed by concate-

nating the coefficient matrix and constant vector.

And we may apply operations known as the Elementary Row Operation.

We will not examine that process.

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Homogeneous Matrix Later, we will use 4x4 matrices to consistently represent

translation, scaling and rotation in 3D space.

In that case a14, a24 and a34 is always zero, so we call this 4x4 matrix as a Homogeneous Matrix.

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Matrix Inverses An n × n matrix M is invertible if there exists a matrix,

which we denote by M−1, such that MM−1 =M−1M = I. The matrix M−1 is called the inverse of M.

Not every matrix has an inverse, and those that do not are called singular. An example of a singular matrix is any one that has a row or column consisting of all zeros.

Any matrix possessing a row that is a linear combination of the other rows of the matrix is singular.

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How to calculate the inverse of an n×n matrix M?

There is a well known algorithm called 'Gauss-Jordan elimination'.

We will not examine that algorithm.

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Back to the 2D Rotation Do you remember this linear system?

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x' = x·cos(θ) – y·sin(θ)y' = x·sin(θ) + x·cos(θ)

Above linear system can be represented like below.=

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Key observation In 2-dimensional space, when a position (a,b) is given, a

linear system uniquely determines a new position (a', b').

For example, for a position (2,1), We can find a linear system which transform to a new position (1,3).

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But there may be no linear system such like that, or may be very difficult to find the system.

In that case, the process may be decomposed to more easier steps.

To consistently represent this process, we uses matrix as a ADT(abstract data type).

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References https://en.wikipedia.org/wiki/Linear_function http://mathworld.wolfram.com/Point-PlaneDistance.html https://en.wikipedia.org/wiki/Matrix_(mathematics)

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