6.837 Linear Algebra ReviewOverviewAdditional ResourcesWhat is a Matrix?Basic OperationsSlide 6Slide 7MultiplicationVector OperationsVector InterpretationVectors: Dot ProductSlide 12Vectors: Cross ProductSlide 14Inverse of a MatrixDeterminant of a MatrixSlide 17Slide 18Homogeneous MatricesOrthonormal BasisChange of Orthonormal BasisSlide 22Slide 23Slide 24Slide 25Slide 26Slide 27CaveatsQuestions?6.837 Linear Algebra Review6.837 Linear Algebra Review6.837 Linear Algebra 6.837 Linear Algebra ReviewReviewRob JagnowMonday, September 20, 20046.837 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 columns11100100aaaan columnsm rowsm×n matrix6.837 Linear Algebra Review6.837 Linear Algebra ReviewBasic OperationsBasic Operations•Transpose: Swap rows with columnsihgfedcbaMifchebgdaMTzyxV zyxVT6.837 Linear Algebra Review6.837 Linear Algebra ReviewBasic OperationsBasic Operations•Addition and SubtractionhdgcfbeahgfedcbahdgcfbeahgfedcbaJust add elementsJust subtract elementsABABA+BAA-B-BA-BAAB6.837 Linear Algebra Review6.837 Linear Algebra ReviewBasic OperationsBasic Operations•MultiplicationdhcfdgcebhafbgaehgfedcbaMultiply each row by each columnAn m×n can be multiplied by an n×p matrix to yield an m×p result6.837 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: n×1 matrix•Interpretation: a point or line in n-dimensional space•Dot Product, Cross Product, and Magnitude defined on vectors onlycbavxyv6.837 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 linesyxdcbayx''VV’6.837 Linear Algebra Review6.837 Linear Algebra ReviewVectors: Dot ProductVectors: Dot Product•Interpretation: the dot product measures to what degree two vectors are alignedABABA=BIf A and B have length 1…A·B = 0 A·B = 1A·B = cos θθ6.837 Linear Algebra Review6.837 Linear Algebra ReviewVectors: Dot ProductVectors: Dot Product cfbeadfedcbaABBATccbbaaAAAT2)cos(BABA Think of the dot product as a matrix multiplicationThe magnitude is the dot product of a vector with itselfThe dot product is also related to the angle between the two vectors6.837 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 sin of the angle between A and B•The direction of C follows the right hand rule if we are working in a right-handed coordinate system)sin(BABA BAA×B6.837 Linear Algebra Review6.837 Linear Algebra ReviewVectors: Cross ProductVectors: Cross ProductThe cross-product can be computed as a specially constructed determinantABA×BxyyxzxxzyzzyzyxzyxbababababababbbaaakjiBAˆˆˆ6.837 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 property100010001I6.837 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 18.06dcbaAbcadA )det(acbdbcadA116.837 Linear Algebra Review6.837 Linear Algebra ReviewDeterminant of a MatrixDeterminant of a MatrixcegbdiafhcdhbfgaeiihgfedcbaihgfedcbaihgfedcbaihgfedcbaFor a 3×3 matrix:Sum from left to rightSubtract from right to leftNote: In the general case, the determinant has n! terms6.837 Linear Algebra Review6.837 Linear Algebra ReviewInverse of a MatrixInverse of a Matrix100010001ihgfedcba1. 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 dimension110001'''222120121110020100zyxtaaataaataaazyxzyx6.837 Linear Algebra Review6.837 Linear Algebra ReviewOrthonormal BasisOrthonormal Basis•Basis: a space is totally defined by a set of vectors – any point is a linear combination of the basis•Orthogonal: dot product is zero•Normal: magnitude is one•Orthonormal: orthogonal + normal•Most common Example: zyxˆ,ˆ,ˆ6.837 Linear Algebra Review6.837 Linear Algebra ReviewChange of Orthonormal Change of Orthonormal BasisBasis•Given: coordinate frames xyz and uvnpoint p = (px, py, pz) •Find:p = (pu, pv,
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