linear-algebra

# linear-algebra - ChrisRambicure GuojinChen ChristopherCprek...

This preview shows pages 1–19. Sign up to view the full content.

Numerical Linear Algebra Chris Rambicure Guojin Chen Christopher Cprek

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
WHY USE LINEAR  ALGEBRA? 1) Because it is applicable in many  problems…. 2)…And it’s usually easier than calculus
TRUE “Linear algebra has become as basic and as applicable as calculus,and fortunately it is easier.” -Gilbert Strang Calculus

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
HERE COME THE BASICS…
SCALARS What you’re used to dealing with Have magnitude, but no direction

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
VECTORS Represent both a magnitude and a  direction Can add or subtract, multiply by scalars,  or do dot or cross products
THE MATRIX It’s an mxn array Holds a set of numerical values Especially useful in solving certain types  of equations Operations: Transpose, Scalar Multiply,  Matrix Add, Matrix Multiply

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
EIGENVALUES You can choose a matrix A, a vector x,  and a scalar x so that Ax = sx, meaning  the matrix just scales the vector X in this case is called an eigenvector,  and s is its eigenvalue
CHARACTERISTIC  EQUATION det(M-tI) = 0 M: the matrix I: the identity t: eigenvalues

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CAYLEY-HAMILTON  THEOREM IF  AND THEN p(A) = 0, meaning A satisfies its  characteristic equation
A Couple Names, A Couple  Algorithms

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
IN THE BEGINNING… (Grassmann’s Linear Algebra) Grassmann is considered to be the  “father” of linear algebra Developed the idea of a linear algebra  in which the symbols representing  geometric objects can be manipulated Several of his operations: the interior  product, the exterior product, and the  multiproduct
What’s a Multiproduct  Equation Look Like? δ 1 ⊗δ 2  +  δ 1 2  = 0 The multiproduct has many uses,  including scientific, mathematic, and  industrial Got updated by William Clifford

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
TO GRASSMAN’S  EQUATION δ 1 ⊗δ 2  +  δ 1 2  = 2k ij The 2k ij  is what’s referred to as  Kronecker’s Symbol Both of these equations are used for  Quantum Theory Math
VECTOR SPACE Another idea which is kind of tied with  Grassman Vector Space refers to some set of  vectors that contains the origin It is usually infinite Subspace is a subset of vector space.   It, of course, is also vector space

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Cholesky Decomposition Algorithm developed by Arthur Cayley Takes a matrix and factors it into a  triangular matrix times its transpose A=R’R Useful for matrix applications Becomes even more worthwhile in  parallel
HOW TO USE LINEAR  ALGEBRA FOR PDE’S You can use matrices and vectors to  solve partial differential equations For equations with lots of variables,  you’ll wind up with really sparse  matrices Hence, the project we’ve been working  on all year

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
BIBLIOGRAPHY “Hermann Grassmann.” Online.  http://members.fortunecity.com/johnhays/grassmann.h
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 65

linear-algebra - ChrisRambicure GuojinChen ChristopherCprek...

This preview shows document pages 1 - 19. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online