1 Last time: row reduction to (reduced) echelon form

Transcription

1 MATH Linear algebra (Fall 8) Lecture Last time: row reduction to (reduced) echelon fm The leading entry in a nonzero row of a matrix is the first nonzero entry from left going right example, the row 7 has leading entry 7, which occurs in the rd column F Definition A matrix with m rows and n columns is in echelon fm if it has the following properties: If a row is nonzero, then every row above it is also nonzero The leading entry in a nonzero row is in a column to the right of the leading entry in the row above If a row is nonzero, then every entry below its leading entry in the same column is zero F example, is in echelon fm, but none of is in echelon fm Definition A matrix is in reduced echelon fm if The matrix is in echelon fm Each nonzero row has leading entry (*) 4 The leading in each nonzero row is the only nonzero number in its column The matrix / / is in reduced echelon fm and is row equivalent to the matrix (*) Theem Each matrix A is row equivalent to exactly one matrix in reduced echelon fm We denote this matrix by RREF(A) The row reduction algithm is a way of constructing RREF(A) from A This algithm is something you should memize and be able to perfm quickly We won t review the full definition again in this lecture, but let s do an example Example Writing to indicate a sequence of row operations, we have and the last matrix is the reduced echelon fm of the first matrix Remark There is another way to think about what this computation means Note that the first matrix is the augmented matrix of the linear system a + b + c a + b + c a + b + c

2 MATH Linear algebra (Fall 8) Lecture where we are now using a, b, c as variables rather than x, x, x as usual A solution to this linear system gives the coefficients of a polynomial f(x) a + bx + cx with f(), f(), and f() The graph of this function is a parabola passing through the points (, ), (, ), and (, ) But these three points are all on the same line y x We therefe must have f(x) x and (a, b, c) (,, ) must be the unique solution to our system This fces the reduced echelon fm of our augmented matrix to be what we computed A pivot column of a matrix A is a column containing a leading in RREF(A) If A is the augmented matrix of a linear system in variables x, x,, x n, then we say that x i is a basic variable if i is a pivot column and that x i is a free variable if i is not a pivot column To determine the basic and free variables of the system, we have to perfm the row reduction algithm to figure out what RREF(A) is first Once we have done this, we can conclude that: The system has solutions if the last column is a pivot column of A The system has solutions if the last column is not a pivot column but there is free variable The system has solution if there are no free variables, and the last column is not a pivot column Meover, here s how you find all the solutions to the system: choose any values f the free variables, then solve f the basic variables in terms of the free variables via the equations which make up the linear system cresponding to RREF(A) Vects Until we see vect spaces later in this course, the term vect will always refer to an dered list of numbers in R A vect (sometimes to be called a column vect) is such a list iented vertically; in other wds, a matrix with one column: 7 6 We write a general column vect as v v v where each v i is a real number Two vects u and v are equal if they have the same number of rows and the same entries in each row The sum of two vects is v n v u u + v v + u u + v v n u n u n + v n Note: u + v v + u, but we can only add together vects with the same number of rows

3 MATH Linear algebra (Fall 8) Lecture If v is a vect and c R is a scalar, ie, a real number, then we define v cv v cv c cv v n cv n We call the new vect cv the scalar multiple of v by c Example We have and + Define subtraction of vects as addition after multiplying by the scalar : + ( ) + We write R n f the set of all vects with exactly n rows Vects a R a can be identified with arrows in the Cartesian plane from the igin to the point (x, y) (a, a ): a 7 Proposition The sum a + b of two vects a, b R is the vect represented by the arrow from the igin to the point which is the opposite vertex of the parallelogram with sides a and b: Proof We have a a (a+b) b (a +b ) b and b b (a+b) a (a +b ) a The fractions a a and b b are the slopes of the lines through the igin containing the vects a and b The other two fractions are the slopes of the lines () between the endpoints of b and a + b and () between the endpoints of a and a + b The first line of the proof shows that line () is parallel to a, and line () is parallel to b Therefe lines () and () are the other two sides of the unique parallelogram with sides a and b

4 MATH Linear algebra (Fall 8) Lecture The endpoint of a + b is where lines () and () intersect Therefe this endpoint is the vertex of the parallelogram opposite the igin The zero vect R n is the vect whose entries are all zero We have + v v + v f any vect v Definition Suppose v, v,, v p R n are vects and c, c,, c p R are scalars, ie, numbers The vect y c v + c v + + c p v p is called a linear combination of v, v,, v p It is the linear combination of v, v,, v p with coefficients c, c,, c p Example Suppose a and b 6 and c If it were, we could find numbers x, x R such that x a + x b c, ie, such that x + x 7 x + x 4 x + 6x So to answer our question we need to determine if this linear system has a solution To do this, use row reduction: 7 7 A Is c a linear combination of a and b? 7 RREF(A) The pivot columns of A are and : the last column is not a pivot column Therefe our linear system is consistent, which means that c is a linear combination of a and b We generalize this example with the following statement Proposition A vect equation of the fm x a + x a + + x n a n b where x, x,, x n are variables and a, a,, a n, b R m are vects, has the same solutions as those f the linear system with augmented matrix a a a a n b (*) This notation means the matrix whose ith column is a i and last column is b In other wds, the vect b is a linear combination of a, a,, a n if and only if the linear system whose augmented matrix is (*) is consistent Definition The span of a vects v, v,, v p R n is the set of all vects y R n that are linear combinations of v, v,, v p We denote the span of some set of vects by What does R-span{v, v,, v p } look like? R-span{v, v,, v p } span{v, v,, v p } We can visualize the span of the vect as the single point consisting of just the igin We imagine the span of a collection of vects that all belong to the same line through the igin as that line In R, if the span of v, v,, v p does not consist of a line, then the span is the whole plane 4

5 MATH Linear algebra (Fall 8) Lecture To see this, imagine we have two vects u, v R which are not parallel We can then get to any point in the plane by travelling some distance in the u direction, then some distance in the v direction In other wds, we can get any vect in R as the linear combination au + bv f some scalars a, b R Draw a picture to illustrate this to yourself:

6 MATH Linear algebra (Fall 8) Lecture Vocabulary Keywds from today s lecture: Vect A vertical list of numbers Equivalently, a matrix with one column The set of all vects with n rows is written R n Example: 4 π Scalar Another wd f number constant We can multiply scalars together, but not vects Example: π The zero vect R n The vect with n rows all equal to zero 4 Linear combination of vects u v u + v If u and v are vects, then u + v u v u + v cv If c R is a scalar then cv cv The linear combination of vects v, v,, v p R n with coefficients a, a,, a p R is Example: 4 + π a v + a v + + a p v p R n + π 8 + π + π 7 + π The span of a list of vects v, v,, v p R n The set of all linear combinations of the vects v, v,, v p R n A vect u R n belongs to the span of v, v,, v p R n if and only if the n (p + ) matrix A v v v p u is the augmented matrix of a consistent linear system This happens precisely when A has no pivot positions in the last column 6