Linear Transformations and Matrices

1. Linear Transformationsand Matrices Jim Van Verth (jim@essentialmath.com) Lars M. Bishop (lars@essentialmath.com)

2. Transformation • Have some geometric data • How to apply functions to it? • Also desired: combine multiple steps into single operation • For vectors: linear transformations Essential Math for Games

3. Transformations • A transformationT:VW is a function that maps elements from vector space V to W • The function f(x,y) = x2 + 2y is a transformation because it maps R2into R Essential Math for Games

4. Linear Transformation • Two basic properties: • T(x + y) = T(x) + T(y) • T(ax) = aT(x) • Follows that • T(0) = 0 • T(ax+y) = aT(x) + T(y) Essential Math for Games

5. Linear Transformations • Basis vectors span vector space • Know where basis goes, know where rest goes • So we can do the following: • Transform basis • Store as columns in a matrix • Use matrix to perform linear transforms Essential Math for Games

6. Linear Transforms • Example: • (1,0) maps to (1,2) • (0,1) maps to (2,1) • Matrix is Essential Math for Games

7. What is a Matrix? • Rectangular m x n array of numbers • M rows by n columns • If n=m, matrix is square Essential Math for Games

8. Matrix Concepts • Number at row i and column j of matrix A is elementAij • Elements in row i make row vector • Elems in column j make column vector • If at least one Aii (diagonal from upper left to lower right) are non-zero and all others are zero, is diagonal matrix Essential Math for Games

9. Transpose • Represented by AT • Swap rows and columns along diagonal • ATij = Aji • Diagonal is invariant Essential Math for Games

10. Transpose • Transpose swaps transformed basis vectors from columns to rows • Useful identity Essential Math for Games

11. Transforming Vectors • Represent vector as matrix with one column • # of components = columns in matrix • Take dot product of vector w/each row • Store results in new vector Essential Math for Games

12. Transforming Vectors • Example: 2D vector • Example: 3D vector to 2D vector Essential Math for Games

13. Row Vectors • Can also use row vectors • Transformed basis stored as rows • Dot product with columns • Pre-multiply instead of post-multiply • If column default, represent row vector by vT Essential Math for Games

14. Row vs. Column • Using column vectors, others use row vectors • Keep your order straight! • Transpose to convert from row to column (and vice versa) Column vector order (us, OpenGL) Row vector order (DirectX) Essential Math for Games

15. Matrix Product • Want to combine transforms • What matrix represents ? • Idea: • Columns of matrix for S are xformed basis • Transform again by T Essential Math for Games

16. Matrix Product • In general, element ABij is dot product of row i from A and column j from B or Essential Math for Games

17. Matrix product (cont’d) • Number of rows in A must equal number of columns in B • Generally not commutative • Is associative Essential Math for Games

18. Block Matrices • Can represent matrix with submatrices • Product of block matrix contains sums of products of submatrices Essential Math for Games

19. Identity • Identity matrix I is square matrix with main diagonal of all 1s • Multiplying by I has no effect • A  I = A Essential Math for Games

20. Inverse • A-1 is inverse of matrix A such that • A-1 reverses what A does • A is orthogonal if AT = A-1 • Component vectors are at right angles and unit length • I.e. orthonormal basis Essential Math for Games

21. Computing Inverse • Only square matrices have inverse • Inverse doesn’t always exist • Zero row, column means no inverse • Use Gaussian elimination or Cramer’s rule (see references) Essential Math for Games

22. Computing Inverses • Most interactive apps avoid ever computing a general inverse • Properties of the matrices used in most apps can simplify inverse • If you know the underlying structure of the matrix, you can use the following: Essential Math for Games

23. Computing Inverse • If orthogonal, A-1 =AT • Inverse of diagonal matrix is diagonal matrix with A-1ii = 1/Aii • If know underlying structure can use • We’ll use this to avoid explicit inverses Essential Math for Games

24. Storage Format • Row major • Stored in order of rows • Used by DirectX Essential Math for Games

25. Storage Format (cont’d) • Column Major Order • Stored in order of columns • Used by OpenGL, and us Essential Math for Games

26. Storage Format (cont’d) • Note: storage format not the same as multiplying by row vector • Same memory footprint: • Matrix for multiplying column vectors in column major format • Matrix for multiplying row vectors in row major format • I.e. two transposes return same matrix Essential Math for Games

27. System of Linear Equations • Define system of m linear equations with n unknowns b1 = a11x1 + a12x2 + … + a1nxn b2 = a21x1 + a22x2 + … + a2nxn … bm = am1x1 + am2 x2+ … + amn xn Essential Math for Games

28. System of Linear Equations • Matrix multiplication encapsulates linear system Essential Math for Games

29. System of Linear Equations • Matrix multiplication encapsulates linear system • Solve for x • Multiplying by inverse imprecise • Use Gaussian elimination, or other methods (see refs) Essential Math for Games

30. References • Anton, Howard and Chris Rorres, Elementary Linear Algebra, 7th Ed, Wiley & Sons, 1994. • Axler, Sheldon, Linear Algebra Done Right, Springer Verlag, 1997. • Blinn, Jim, Notation, Notation, Notation, Morgan Kaufmann, 2002. Essential Math for Games