Lecture 7 R pivot columns have a single 1 at the pivot Rank number of pivots Rank is never greater than the number of rows Solving Ax 0 leads us to Rx 0 Supposed we have a 5x6 matrix A the nullspace is greater than just the origin There at most 5 pivot columns meaning that one column is free This free column allows us to construct a special solution We take 1 for the free variable Linear equation y 3y 2y e Linear equation x x 6 There are in nitely many particular solutions The null equation x x 0 All solutions x x Ax b Take A to R and do the same to b all in one matrix If there is a zero in row n of R there must be a zero in row n of d If this is not true there are no solutions Choosing a particular solution Put zeros in for the free variables and solve the equations All solutions Example Look at the null solution because all solutions are given by all null solutions plus one particular solution
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