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Review for Final

by: Mae Underwood

Review for Final MAT22A

Mae Underwood
GPA 3.4
Linear Algebra
R. Granero Belinchon

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About this Document

Summary of concepts and how to approach each problem you may encounter on the final
Linear Algebra
R. Granero Belinchon
Study Guide
50 ?




Popular in Linear Algebra

Popular in Mathematics (M)

This 6 page Study Guide was uploaded by Mae Underwood on Sunday December 6, 2015. The Study Guide belongs to MAT22A at University of California - Davis taught by R. Granero Belinchon in Fall 2015. Since its upload, it has received 33 views. For similar materials see Linear Algebra in Mathematics (M) at University of California - Davis.

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Date Created: 12/06/15
Linearly dependent vectors Vectors are linear combinations of each other Linearly independent vectors Linear combination c117 c217 0 gives only the trivial solution c1 c2 0 Given a set of vectors let A be the matrix whose columns are formed by these vectors A v1 vZ then RREFA If A is a square matrix it is only linearly independent IOI A is invertible detA O Span o A linear span of a set of vectors a vector space The span of any set of vectors is a subspace o Intersection of all subspaces containing that set of vectors 0 To check which subspace s spans find rankA o If two 2D vectors are linearly dependent their span is a line 0 To check if a set is in a vector space does the set contain the zero vector If yes check if set is closed under scalar multiplication and addition 0 Set of all vectors that can be created by their linear combination 0 if the scalar multipliers are nonzero then the set is linearly dependent and the matrix formed by these vectors is singular o If scalars are ALL zeroes set is linearly independent and matrix is invertible Basis Rules Vectors are linearly independent and they span S Row space RA Vector space spanned by rows of Amxn Is a subspace of 1R Set of all linear combinations of row vectors of A RREFA nonzero rows form a basis for RA RA CAT Column space CA Spanned by columns of Amxn Is a subspace of R39 Set of all linear combinations of columns of A Set of all vectors 3 such that A I is consistent A RREFA find columns of matrix A that correspond to the pivot columns of rrefA B CA RAT Check 1 Show that ABJE I is consistent find rrefAB E and solve 2 RankAB RankABI E RankA RankA Dim RA no of vectors in a basis for RA RankA Dim CA no of linearly independent rows f columns of matrix A is formed by a set of vectors S then RankA n where S spans 1R Row operations change CA but not RA 9 so arrange vectors in rows to form matrix A Nullspace NA Set of all solutions of the linear system A 6 Subspace of 1R Amxn Nullity dimension of the Nullspace no of free variables Left Nullspace MN Find Nullspace basis 1 2 RREFA and solve for leading variables Identify nonpivot columns as free variables find span of Nullspace using these solutions Dimension of a vector space minimum number of coordinates needed to specify any vector within the VECtOl39 space Let Am x n RankA Dim RA Dim CA Dim CAT If subspace of RA is 1R rankA n RankA NullityA n n columns of A RankAT NullityAT m m rows of A Contained in Subspace Orthogonal to NA RA CAT 1R columns CAT NA 1R columns NAT CA Rm rows CA NAT Rm rows Basis Rm vectors in basis of MA and CAT RA Eigenvalues and Eigenvectors Let A M B All nonzero solutions of detB 0 are eigenvalues An detA AI gt quotcharacteristic polynomial of A All nonzero solutions of B O are eigenvectors in 0 When RREFBO expect to find at least one zero row predetermined by setting detBO Orthonormal Basis P9P Vectors are unitary and orthogonal between them Check that the set of vectors form a basis linear independence Check if and 17 from the set are orthogonal if not let 1i 1 Proj Do the same for the rest of the set Wi W Proj w Proj iW Normalize all the orthogonal vectors to get a magnitude 1 for each Orthogonal Basis 1 Follow the same procedure as above but don t normalize 1747 Pr0117u 2v 1717 17 2 A d a T Hall 1717 Il llll ll c056 c056 S 1 Finding the determinate detA A GE to upper triangular form U and take product of the main diagonal B 2219 Cl ja CIE l 9 def 6f df de o DetA detAT o DetA B at detA detB o DetAB detAdetB o DetA changes sign when a rowcolumn is exchanged o If 2 rows of A are equal detA 0 o DetA does not change with row operations except row exchanges 1 DetA 1 detA o The determinate is a linear function of a given rowcolumn Finding the Solution to A b A RREF gt get as close as possible to identity matrix I 1 Select row pivots and eliminate everything else in column 2 Optional turn pivots main diagonal into ls 3 Solve for 3c B Gaussian Elimination GE to upper triangular form then solve resulting system of equa ons C LDU Factorization A can only reduced to U by GE without exchanging rows 1 A LU 2 A22132LU22B 3 Let U27 solve L37 2 I3 4 Solve 37 U for a a b c Find U through GE to obtain 0 e f O O 139 Find L by starting with identity matrix I 0 Each time a row operation is performed for U replace the negative of the multiplier in the corresponding space 1 b c R2 kR1 gt k 1 f 001 o Diagonals should be ls but if the row operation involves turning the main diagonal of U to ls put lk where the leading 1 occurs 1 b C Le kRz gt 0 1k f 0 0 1 D Inverse Matrix 1 Find out if A has an inverse detA 0 2 Find inverse of A A L 31422213232214413 Finding inverse A fA is a 2 x 2 matrix 1 19 A l detlm d b d c a a b c 1 0 0 B Start with this augmented matrix d e f O 1 0 g h i 0 0 1 1 Perform matrix operations until we get I on the left side 100jkl Olomno 001pqr k l 2 A 1 m n o p q 7quot 3 Check AA L Complete Solution Rank RREFAI 5 Particular solution Nullspace General solution SquotPP Nquot o If there unknowns at equations there are many solutions not linearly independent 0 No free variables dimension of vector space formed by solutions 0 If finding Nullspace nonpivot columns free variables nullity o If A 13 has no solution multiple each side by AT to get a square matrix ATAFC ATE Pivots first nonzero coefficient in each row Free variables nonpivot columns after RREFA only exists if there are many solutions not linearly independen a Find Projection Matrix of Line in direction of E b c Find a Observe dimension of d if in 1R3 then start with a 3 x 3 identity matrix Each column of the identity matrix must be transformed let each column be x1 x2 and x3 If the projection matrix is P with columns p1 p2 and p3 P9P pn x11 39 aa Projections are the best possible approximation inside a certain subspace V


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