matrices review test 1

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

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

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

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

View Full Document
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
Ask a homework question - tutors are online