boot camp handout - Row-Reduction Boot Camp Constantin...

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Constantin Teleman, Math 54.1, Fall 09 1. Notation and refreshers Unless otherwise specified, A is an m × n real matrix, and vectors x are column vectors. Row vectors are written as x T , where x is the corresponding column vector. The nullspace of A , Null( A ) R n , is the space of solutions of the homogeneous system A x = 0 , the column space Col( A ) R m is the span of the columns. There are two more subspaces associated to A , this time consisting of row vectors : the row space Row( A ) R n , the span of the rows in A ; and the left nullspace LNull( A ) R m that we’ll meet below. (A quick and dirty definition is as the nullspace of the ‘flipped’ or transposed matrix A T : this is the matrix with switched indexing, ( A T ) ij = A ji .) Caution! These four subspaces are distinct in general, with no obvious relation among them. We will learn some subtle relations later. The reduced row echelon form of A is the matrix rref( A ) produced from elementary row oper- ations, with the properties that All zero-rows are at the bottom Every row which is not all zero starts with a 1, called the pivot or leading 1, Every pivot is strictly to the right of all pivots in the rows above it, and All entries above (and below) a pivot are zero. The columns containing pivots are called pivot columns , the others are the free columns . The nullspace of A agrees with that of rref( A ), and can be parametrized as follows: the free variables can be chosen freely, and each equation in the reduced row system can then be used to solve for the corresponding pivot variable. A similar story applies to the inhomogeneous system A x = b : it has the same general solution as the reduced system rref( A ) x = b 0 , where [rref( A ) | b 0 ] is the reduced form of the augmented matrix [ A | b ]. The general solution can be parametrized by the same procedure. We will learn a strong uniqueness property of rref( A ): it is completely determined by nullspace of A . (Similarly, the reduced form of the augmented matrix [ A | b ] is determined uniquely from the affine space of solutions of A x = b .) A collection of vectors { v 1 ,..., v r } is linearly independent if the only expression of 0 as a linear combination of the v i is the one with 0 weights: that is, k z v 1 + ··· + k r v r = 0 k 1 = k 2 = ··· = k r = 0 . A collection { w 1 ,..., w s } spans the linear subspace L R n if every w i lies in L and every vector in L can be expressed as a linear combination of the w i . A linearly independent, ordered collection of vectors which spans L is called a basis . Each vector in L can be uniquely expressed as a linear combination of the basis elements (that is, the weights are uniquely determined). The main example is the standard basis e 1 ,..., e n of R n , the unit vectors on the coordinate axes. 1
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This note was uploaded on 12/01/2009 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at University of California, Berkeley.

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boot camp handout - Row-Reduction Boot Camp Constantin...

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