study methodFor each lecture:Watch the lecture video.Don’t try to copy every word.Pause only when a derivation is confusing.Read the lecture notes (10–20 minutes).Highlight important definitions and theorems.Write a short summary in your own notes.Work through the example problems.Try each one yourself before looking at the solution.Try the problem set.This is the most important part.If you get stuck, go back to the lecture or examples.Watch the recitation only ifseveral homework problems were difficult, oryou want more examples before moving on.Use the mathlet if you’re having trouble visualizing the concept.They’re especially helpful for limits, derivatives, optimization, and integration.indexGeometric interpretation of derivatives 1Notations of derivatives 1Some basic limits/derivatives of functions 1Proof for differentiable - continuous, example for continuous but not differentiable 2Conditions for differentiable 2Make the piecewise function differentiable (smoothing) 2dsin(x)dx\frac{dsin(x)}{dx}dxdsin(x)​algebraic derivation 3limθ→0sin(θ)θ1\underset{\theta\rarr0}{lim}\frac{sin(\theta)}{\theta}1θ→0lim​θsin(θ)​1,limθ→01−cos(θ)θ0\underset{\theta\rarr0}{lim}\frac{1-cos(\theta)}{\theta}0θ→0lim​θ1−cos(θ)​0geometric proof 4dsin(x)dxcos(θ)\frac{dsin(x)}{dx}cos(\theta)dxdsin(x)​cos(θ)geometric proof 5product rule, quotient rule 5creation of the graph ofysin(x)xy\frac{sin(x)}{x}yxsin(x)​6proof for product rule, proof for quotient rule 6composition and chain rule 7notation for higher derivatives 7DnxnD^nx^nDnxnn-th derivative of x to the n-th power 8antiderivative 8transform limits into standard derivatives to evaluate limits 9vectors span a line, a plane, or the 3D space 9invertible×\times×linear equations 10row operations by matrix multiplication 10put all row operations into one matrix 11permutation matrix (exchange rows) 12inverse by row operations 120 in the pivot position 12proof of why Gaussian elimination works for finding inverse 125 ways of treating matrix multiplication 13matrix inverse interpretation of invertibility 15block elimination 14inverse ofABABAB15(AT)−1(A−1)T(A^T)^{-1}(A^{-1})^{T}(AT)−1(A−1)T15LULULUdecomposition 16permutation matrix 17transpose 17vector spaces 17property about dot product(Ax)TyxT(ATy)(Ax)^T yx^T(A^T y)(Ax)TyxT(ATy)18is the union / intersection of two subspaces still a subspace? 18the solutions to Ax0 always give us a subspace? 18pivot column, free column special solution ( finding the null space ) 19how elimination affect vector spaces formed by a matrix 20reduced row echelon form (RREF) how to use it to quickly find null space 20check solvability of b with row echelon form (REF) 21how reduced row echelon form (RREF) tells us the number of solutions to a linear equation 22independence 22basis 23dimension of column space, dimension of null space, and rank of a matrix 23four fundamental subspaces 23why row operations during elimination tell us the independent columns in the original matrix 24find the left null space with elimination 24vector space that contains matrices (matrix space) 25rank one matrix outer product of two vectors ; rank two matrix rank one matrix rank one matrix 26dim(S)dim(U)dim(SU)dim(S inter U) 26graphs and incidence matrices, potential and current 27Euler’s formula 28interpretation of and the way to find the four subspaces of incidence matrix 29orthogonal vectors, orthogonal bases, orthogonal subspaces 30row space is orthogonal to null space, column space is orthogonal to left null space 30solve Axb when b is not in the column space of A (approximation, find the best fit with a small number of parameters) 31properties aboutATAA^T AATA, proof forrank(ATA)rank(A)rank(A^T A)rank(A)rank(ATA)rank(A)31requirements on dimensions for two subspaces to be orthogonal dim(V)dim(W)dim(whole space) 32definition of orthogonal complement 33every vector in the row space produces a unique vector in the column space (one to one relationship) 33every matrix with rank r have a r*r invertible matrix inside it 34property about basis (when the number of vectors is correct, two way convertion: independence↔\leftrightarrow↔span the whole space) 35Fredholm’s Alternative (exactly one of these has solution: Axb,ATy0A^Ty0ATy0andyTb1y^Tb1yTb1) 35projection 35usage of projection in approximation (solve Axb when there is no solution) 36formula of finding the best solution for any Axb 37avoid calculating projection matrix to a high dimensional subspace. Vector projection onto high dimensional subspace projection onto lower dimensional subspace. projection matrix onto high dimensional subspace I - projection matrix onto lower dimensional subspace 37special cases of projection when solving Axb 38geometric interpretation of projecting onto the column space and left null space 39least squares / linear regression 39invertibility of A^T A 40orthonormal vectors 40orthonormal matrices 40Gram-Schmidt 41projection onto orthonormal matrices (rectangular) 43usage of Gram-Schmidt (solve Axb) 4310 properties of determinant 43deriving the cofactor formula for determinant 46visual way to do 3 by 3 determinant 48determinant of tridiagonal matrix 48inverse of A with cofactors 48Cramer’s rule 49find the pivots from determinant 49determinant of pascal matrix 49the absolute value of determinant equals to the volume of a box 50cross product 53eigenvectors eigenvalues 53eigenvalues, eigenvectors of projection matrix, reflection matrix,rotation matrix(need proof),symmetric matrix (need proof),triangular matrix, singular matrix 54-55eigenvalue and eigenvector ofA2A^2A2,A−1A^{-1}A−1,AkIAkIAkI54-55relationship between eigenvalues and rank of a matrix 55geometric and algebraic multiplicity of eigenvalues 56(need proof on inequality)diagonalization / eigendecomposition 57similar matrices 59differential equation 60(incomplete section)implicit differentiation 63differentiating exponentials logarithms 64natural log 65-67derive the formula of the derivative ofxrx^rxrfor any real numberrrrwitheee67hyperbolic trig functions 67evaluating dy/dx when there are other variables 68linear approximation 69quadratic approximation 70linear approximation of f(x)g(x) linear approximation of f(x) times linear approximation o f g(x) 70usage of linear approximation when a function is defined implicitly 70real life example of linear approximation: time dilation (relative error) 71explanation of the quadratic term in the formula of quadratic approximation 71quadratic approximation continued 72curve sketching 72general strategy of graphing 73maxima and minima 75related rates 76Newton’s method 79mean value theorem (MVT) 80application of mean value theorem in graphing 81mean value theorem vs. linear approximation 82how MVT helps with approximations 82generalization of MVT: Taylor’s Theorem 83where the mysterious 2!, 3!, …, n! come from in Taylor’s polynomial 84