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Bishop (lars@essentialmath.com).Nt                    mETransformation    Have some geometric data How to apply functions to it? Also desired: combine multiple steps into single operation For vectors: linear transformations    KTransformations    FA transformation T:VW 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 R2 into R`/3  %  .=  S    `Linear TransformationhTwo 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) % ! ,5# bLinear TransformationsBasis 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&hUhUvLinear Transforms;Example: (1,0) maps to (1,2) (0,1) maps to (2,1) Matrix issWhat is a Matrix?    SRectangular m x n array of numbers M rows by n columns If n=m, matrix is square HT% S    tMatrix Concepts    Number at row i and column j of matrix A is element Aij 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' W T4      !      0    c    d Transpose     \Represented by AT Swap rows and columns along diagonal ATij = Aji Diagonal is invariant] +   \;              8 Transpose     QTranspose swaps transformed basis vectors from columns to rows Useful identity 6PPQ    eTransforming 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    Transforming Vectors    6Example: 2D vector Example: 3D vector to 2D vector 676    k Row VectorsCan 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@ Row vs. Column    Using column vectors, others use row vectors Keep your order straight! Transpose to convert from row to column (and vice versa)d-ZZZZ9Z-9     rMatrix ProductWant to combine transforms What matrix represents ? Idea: Columns of matrix for S are xformed basis Transform again by T bC?C&_yMatrix Product    KIn general, element ABij is dot product of row i from A and column j from B\L 4    3    z.Matrix product (cont d)    _Number of rows in A must equal number of columns in B Generally not commutative Is associative:`!+_    eBlock Matrices    hCan represent matrix with submatrices Product of block matrix contains sums of products of submatricesP       5       CIdentity    Identity matrix I is square matrix with main diagonal of all 1s Multiplying by I has no effect A I = Ac Bl    fInverse    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[M      L   4         gComputing Inverse    6Only square matrices have inverse Inverse doesn t always exist Zero row, column means no inverse Use Gaussian elimination or Cramer s rule (see references)4e     -    ?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:    9Computing Inverse    RIf 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   4   RBT      Q     aStorage Format    6Row major Stored in order of rows Used by DirectX* Z-Z -6    !b.Storage Format (cont d)    JColumn Major Order Stored in order of columns Used by OpenGL, and us *Z8Z8J    "c.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 matrixHVu'>u'@System of Linear Equations    LDefine system of m linear equations with n unknowns b1 = a11x1 + a12 x2 + & + a1n xn b2 = a21x1 + a22 x2 + & + a2n xn & bm = am1x1 + am2 x2+ & + amn xn4s                      vX                   PSystem of Linear Equations     |System of Linear Equations      References     Anton, Howard and Chris Rorres, Elementary Linear Algebra, 7th Ed, Wiley & Sons, 1994. 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