# Math Review Guide.pdf - Math 54 review The following is a...

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Math 54 reviewThe following is a (possibly incomplete rough draft of a) list of keypoints from the class.Making sure you know all of these topics and cansolve problems using them should be good preparation for the final exam.Linear algebra chapter 1.How to solve a system of linear equations by row reduction.Pivotvariables and free variables.How to multiply a matrix by a vector, either by taking dot productswith the rows, or by taking a linear combination of the columns.The definition of linear independence.The definition of a linear transformation.Linear algebra chapter 2.How to add and multiply matrices.Associative and distributive properties of matrix multiplication. Ma-trix multiplication is (usually) not commutative!Equivalence betweenm×nmatrices and linear transformations fromRntoRm. Matrix multiplication corresponds to composition of lineartransformations.IfAis anm×nmatrix, thenTAis injective if and only if the columnsofAare linearly independent, or equivalently every column containsa pivot, which requiresmn; andTAis surjective if and only if thecolumns ofAspanRm, or equivalently every row contains a pivot,which requiresmn.AandBare inverses of each other if bothABandBAequal theidentity.How to test whether a matrix is invertible and how to compute theinverse when it exists, either by row reduction (in the general case) orCramer’s rule (in the 2×2 case).1
Matrices which are not square cannot be invertible. The reason is thatifAis anm×nmatrix andAB=IthenTAis surjective andTBisinjective, both of which requiremn. IfBA=Ialso, then likewisenm.IfAandBaren×nmatrices withAB=I, thenBA=Ialso.Linear algebra chapter 3.How to calculate the determinant of a square matrix, either by rowreduction, by expanding along a row or column, or as a sum overpermutations (at least for 2×2 and 3×3).WhenAis a square matrix,Ais invertible if and only if det(A)6= 0.The geometric meaning of determinants: ifAis ann×nmatrix andUis a region inRnwith finite volume, then vol(TA(U)) =|det(A)|vol(U);and det(A)>0 whenApreserves orientation.Linear algebra chapter 4.Definition of a vector space. Main examples:Rn, subsets of a vectorspace, and some function spaces.Definition of a subspace of a vector space. Examples: span of a subset;

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Term
Spring
Professor
Chorin
Tags
Math, LINEAR ALGEBRA Chapter
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