matrices review test 1

matrices review test 1 - Matrices Notes- Test 1 (M-maybe)...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Matrices Notes- Test 1 (M-maybe) 1.1- Systems of linear equations: o Row operations are reversible o Summary of the elimination method for 1.1:. Check!!! Substitute into original. o Two fundamental Questions about a linear system: Is the system consistent; that is, does at least one solution exist? If a solution exists, is it the only one; that is, is the solution unique? 1.2- Row Reduction and Echelon Forms o The row reduction algorithm applies to ANY matrix, whether it’s augmented or not. o Theorem 1 - Uniqueness of the Reduced Echelon Form: Each matrix is row equivalent to one and only one reduced echelon matrix. o Solving a system amounts to finding a parametric description of the solution set or determining that the solution set is empty. o Theorem 2 - Existence and Uniqueness Theorem: A linear system is consistent if and only if the rightmost column of the augmented matrix is NOT a pivot column—that is, if and only if an echelon form of the augmented matrix has NO row of the form: [0 … 0 b] with b nonzero If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable. o Restatement of last sentence using concept of pivot columns: If a linear system is consistent, then the solution is unique if and only if every column in the coefficient matrix is a pivot column; otherwise, there are infinitely many solutions. o Problems: 1, 5,19,21 (b),22,27M, 31M 1.3- Vector Equations o Parallelogram Rule for addition: if u and v in R 2 are represented as points in the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other vertices are u , 0 , and v . o If v 1 , . . . , v p are in R n , then the set of all linear combinations of v 1 , …, v p is denoted by Span { v 1 , …, v p } and is called the subset of R n spanned (or generated ) by v 1 , …, v p . That is, Span { v 1 , …, v p } is the collection of all vectors that can be written in the form c 1 v 1 + c 2 v 2 + … + c p v p with c 1 , … , c p scalars. So, asking whether a vector b is in Span { v 1 , …, v p } amounts to asking whether the vector equation x 1 v 1 + x 2 v 2 + … + x p v p = b has a solution, or equivalently, asking whether the linear system with augmented matrix [ v 1 . . . v
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/27/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas.

Page1 / 3

matrices review test 1 - Matrices Notes- Test 1 (M-maybe)...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online