6.837 Linear Algebra ReviewOverviewAdditional ResourcesWhat is a Matrix?Basic OperationsMultiplicationVector OperationsVector InterpretationVectors: Dot ProductVectors: Dot ProductVectors: Cross ProductInverse of a MatrixDeterminant of a MatrixDeterminant of a MatrixInverse of a MatrixHomogeneous MatricesOrthonormal BasisOrthonormal BasisOrthonormal BasisQuestions?6.837 Linear Algebra Review6.837 Linear Algebra ReviewPatrick NicholsThursday, September 18, 20036.837 Linear Algebra Review6.837 Linear Algebra ReviewOverviewOverview• Basic matrix operations (+, -, *)• Cross and dot products• Determinants and inverses• Homogeneous coordinates• Orthonormal basis6.837 Linear Algebra Review6.837 Linear Algebra ReviewAdditional ResourcesAdditional Resources• 18.06 Text Book• 6.837 Text Book• [email protected]• Check the course website for a copy of these notes6.837 Linear Algebra Review6.837 Linear Algebra ReviewWhat is a Matrix?What is a Matrix?• A matrix is a set of elements, organized into rows and columns⎥⎦⎤⎢⎣⎡dcbarowscolumns6.837 Linear Algebra Review6.837 Linear Algebra ReviewBasic OperationsBasic Operations• Addition, Subtraction, Multiplication⎥⎦⎤⎢⎣⎡++++=⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡hdgcfbeahgfedcbaJust add elements⎥⎦⎤⎢⎣⎡−−−−=⎥⎦⎤⎢⎣⎡−⎥⎦⎤⎢⎣⎡hdgcfbeahgfedcbaJust subtract elements⎥⎦⎤⎢⎣⎡++++=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡dhcfdgcebhafbgaehgfedcbaMultiply each row by each column6.837 Linear Algebra Review6.837 Linear Algebra ReviewMultiplicationMultiplication• Is AB = BA? Maybe, but maybe not!• Heads up: multiplication is NOT commutative!⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡.........bgaehgfedcba⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡.........fceadcbahgfe6.837 Linear Algebra Review6.837 Linear Algebra ReviewVector OperationsVector Operations• Vector: 1 x N matrix• Interpretation: a line in N dimensional space• Dot Product, Cross Product, and Magnitude defined on vectors only⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=cbavrxyv6.837 Linear Algebra Review6.837 Linear Algebra ReviewVector InterpretationVector Interpretation• Think of a vector as a line in 2D or 3D• Think of a matrix as a transformation on a line or set of lines6.837 Linear Algebra Review6.837 Linear Algebra Review⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡''yxdcbayxVV’Vectors: Dot ProductVectors: Dot Product• Interpretation: the dot product measures to what degree two vectors are alignedABABCA+B = C(use the head-to-tail method to combine vectors)6.837 Linear Algebra Review6.837 Linear Algebra ReviewVectors: Dot ProductVectors: Dot Product[]cfbeadfedcbaabbaT++=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡==⋅Think of the dot product as a matrix multiplicationThe magnitude is the dot product of a vector with itselfccbbaaaaaT++==2The dot product is also related to the angle between the two vectors – but it doesn’t tell us the angle)cos(θbaba =⋅6.837 Linear Algebra Review6.837 Linear Algebra ReviewVectors: Cross ProductVectors: Cross Product• The cross product of vectors A and B is a vector C which is perpendicular to A and B• The magnitude of C is proportional to the cosine of the angle between A and B• The direction of C follows the right hand rule –this why we call it a “right-handed coordinate system”)sin(θbaba =×6.837 Linear Algebra Review6.837 Linear Algebra ReviewInverse of a MatrixInverse of a Matrix• Identity matrix: AI = A• Some matrices have an inverse, such that:AA-1= I• Inversion is tricky:(ABC)-1= C-1B-1A-1Derived from non-commutativity property⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=100010001I6.837 Linear Algebra Review6.837 Linear Algebra ReviewDeterminant of a MatrixDeterminant of a Matrix•Used for inversion• If det(A) = 0, then A has no inverse• Can be found using factorials, pivots, and cofactors!• Lots of interpretations – for more info, take 18.06⎥⎦⎤⎢⎣⎡=dcbaAbcadA−=)det(⎥⎦⎤⎢⎣⎡−−−=−acbdbcadA116.837 Linear Algebra Review6.837 Linear Algebra ReviewDeterminant of a MatrixDeterminant of a Matrixcegbdiafhcdhbfgaeiihgfedcba−−−++=ihgfedcbaihgfedcbaihgfedcbaSum from left to rightSubtract from right to leftNote: N! terms6.837 Linear Algebra Review6.837 Linear Algebra ReviewInverse of a MatrixInverse of a Matrix⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+100010001ihgfedcba1. Append the identity matrix to A2. Subtract multiples of the other rows from the first row to reduce the diagonal element to 13. Transform the identity matrix as you go4. When the original matrix is the identity, the identity has become the inverse!6.837 Linear Algebra Review6.837 Linear Algebra ReviewHomogeneous MatricesHomogeneous Matrices• Problem: how to include translations in transformations (and do perspective transforms)• Solution: add an extra dimension[]⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡111111zyxzyx6.837 Linear Algebra Review6.837 Linear Algebra ReviewOrthonormalOrthonormalBasisBasis• Basis: a space is totally defined by a set of vectors – any point is a linear combinationof the basis• Ortho-Normal: orthogonal + normal• Orthogonal: dot product is zero• Normal: magnitude is one• Example: X, Y, Z (but don’t have to be!)6.837 Linear Algebra Review6.837 Linear Algebra ReviewOrthonormalOrthonormalBasisBasis000=⋅=⋅=⋅zyzxyx[][][]TTTzyx100010001===X, Y, Z is an orthonormal basis. We can describe any 3D point as a linear combination of these vectors.How do we express any point as a combination of a new basis U, V, N, given X, Y, Z?6.837 Linear Algebra Review6.837 Linear Algebra ReviewOrthonormalOrthonormalBasisBasis⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⋅+⋅+⋅⋅+⋅+⋅⋅+⋅+⋅=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡ncnbnavcvbvaucubuanvunvunvucba333222111000000(not an actual formula – just a way of thinking about it)To change a point from one coordinate system to another, compute the dot product of each coordinate row with each of the basis vectors.6.837 Linear Algebra Review6.837 Linear Algebra ReviewQuestions?Questions?6.837 Linear Algebra Review6.837 Linear Algebra
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