Row Space, Col Space, and Rank-Nullity Math 240 Row Space and Column Space Row Space, Column Space, and the The Rank-Nullity Theorem Rank-Nullity Theorem Homogeneous linear systems Nonhomogeneous linear systems Math 240 | Calculus III Summer 2013, Session II Monday, July 22, 2013 Row Space, Col Space, Agenda and Rank-Nullity Math 240 Row Space and Column Space The Rank-Nullity Theorem 1. Row Space and Column Space Homogeneous linear systems Nonhomogeneous linear systems 2. The Rank-Nullity Theorem Homogeneous linear systems Nonhomogeneous linear systems Row Space, Col Space, Motivation and Rank-Nullity Math 240 n Say S is a subspace of R with basis fv1; v2;:::; vng. What Row Space operations can we perform on the basis while preserving its and Column Space span and linear independence? The Rank-Nullity I Swap two elements (or shuffle them in any way) Theorem Homogeneous linear systems E.g. fv1; v2; v3g ! fv2; v1; v3g Nonhomogeneous linear systems I Multiply one element by a nonzero scalar E.g. fv1; v2; v3g ! fv1; 5v2; v3g I Add a scalar multiple of one element to another E.g. fv1; v2; v3g ! fv1; v2; v3 + 2v2g If we make the v1;:::; vn the rows of a matrix, these operations are just the familiar elementary row ops. Row Space, Col Space, Row space and Rank-Nullity Math 240 Row Space and Column Definition Space If A is an m × n matrix with real entries, the row space of A The n Rank-Nullity is the subspace of R spanned by its rows. Theorem Homogeneous linear systems Nonhomogeneous Remarks linear systems 1. Elementary row ops do not change the row space. 2. In general, the rows of a matrix may not be linearly independent. Theorem The nonzero rows of any row-echelon form of A is a basis for its row space. Row Space, Col Space, Example and Rank-Nullity Determine a basis for the row space of Math 240 2 3 Row Space 1 −1 1 3 2 and Column Space 62 −1 1 5 17 A = 6 7 : The 43 −1 1 7 05 Rank-Nullity Theorem 0 1 −1 −1 −3 Homogeneous linear systems Nonhomogeneous linear systems Reduce A to the row-echelon form 21 −1 1 3 23 60 1 −1 −1 −37 6 7 : 40 0 0 0 05 0 0 0 0 0 Therefore, the row space of A is the 2-dimensional subspace of 5 R with basis (1; −1; 1; 3; 2); (0; 1; −1; −1; −3) : Row Space, Col Space, Column space and Rank-Nullity Math 240 Row Space and Column Space The We can do the same thing for columns. Rank-Nullity Theorem Homogeneous Definition linear systems Nonhomogeneous If A is an m × n matrix with real entries, the column space of linear systems m A is the subspace of R spanned by its columns. Obviously, the column space of A equals the row space of AT , so a basis can be computed by reducing AT to row-echelon form. However, this is not the best way. Row Space, Col Space, Example and Rank-Nullity Math 240 Determine a basis for the column space of Row Space 2 3 and Column 1 2 −1 −2 0 Space 62 4 −1 1 07 The A = 6 7 = a1 a2 a3 a4 a5 : Rank-Nullity 43 6 −1 4 15 Theorem Homogeneous 0 0 1 5 0 linear systems Nonhomogeneous linear systems Reduce A to the reduced row-echelon form 21 2 0 3 03 60 0 1 5 07 E = 6 7 = e1 e2 e3 e4 e5 : 40 0 0 0 15 0 0 0 0 0 e2 = 2e1 ) a2 = 2a1 e4 = 3e1 + 5e3 ) a4 = 3a1 + 5a3 Therefore, fa1; a3; a5g is a basis for the column space of A. Row Space, Col Space, Column space and Rank-Nullity Math 240 Row Space and Column We don't need to go all the way to RREF; we can see where Space the leading ones will be just from REF. The Rank-Nullity Theorem Theorem Homogeneous linear systems If A is an m × n matrix with real entries, the set of column Nonhomogeneous linear systems vectors of A corresponding to those columns containing leading ones in any row-echelon form of A is a basis for the column space of A. Another point of view m The column space of an m × n matrix A is the subspace of R m consisting of the vectors v 2 R such that the linear system Ax = v is consistent. Row Space, Col Space, Relation to rank and Rank-Nullity Math 240 Row Space and Column Space If A is an m × n matrix, to determine bases for the row space The and column space of A, we reduce A to a row-echelon form E. Rank-Nullity Theorem Homogeneous 1. The rows of E containing leading ones form a basis for the linear systems Nonhomogeneous row space. linear systems 2. The columns of A corresponding to columns of E with leading ones form a basis for the column space. dim (rowspace(A)) = rank(A) = dim (colspace(A)) Row Space, Col Space, Segue and Rank-Nullity Math 240 If A is an m × n matrix, we noted that in the linear system Row Space and Column Ax = v; Space The Rank-Nullity rank(A), functioning as dim (colspace(A)), represents the Theorem Homogeneous degrees of freedom in v while keeping the system consistent. linear systems Nonhomogeneous linear systems The degrees of freedom in x while keeping v constant is the number of free variables in the system. We know this to be n − rank(A), since rank(A) is the number of bound variables. Freedom in choosing x comes from the null space of A, since if Ax = v and Ay = 0 then A(x + y) = Ax + Ay = v + 0 = v: Hence, the degrees of freedom in x should be equal to dim (nullspace(A)). Row Space, Col Space, The Rank-Nullity Theorem and Rank-Nullity Math 240 Row Space and Column Space Definition The Rank-Nullity When A is an m × n matrix, recall that the null space of A is Theorem Homogeneous n linear systems nullspace(A) = fx 2 : Ax = 0g : Nonhomogeneous R linear systems Its dimension is referred to as the nullity of A. Theorem (Rank-Nullity Theorem) For any m × n matrix A, rank(A) + nullity(A) = n: Row Space, Col Space, Homogeneous linear systems and Rank-Nullity Math 240 Row Space We're now going to examine the geometry of the solution set and Column Space of a linear system. Consider the linear system The Rank-Nullity Ax = b; Theorem Homogeneous linear systems Nonhomogeneous where A is m × n. linear systems If b = 0, the system is called homogeneous. In this case, the solution set is simply the null space of A. Any homogeneous system has the solution x = 0, which is called the trivial solution. Geometrically, this means that the solution set passes through the origin. Furthermore, we have shown that the solution set of a homogeneous system is in fact n a subspace of R . Row Space, Col Space, Structure of a homogeneous solution set and Rank-Nullity Math 240 Row Space and Column Theorem Space The I If rank(A) = n, then Ax = 0 has only the trivial solution Rank-Nullity Theorem x = 0, so nullspace(A) = f0g. Homogeneous linear systems I If rank(A) = r < n, then Ax = 0 has an infinite number Nonhomogeneous linear systems of solutions, all of which are of the form x = c1x1 + c2x2 + ··· + cn−rxn−r; where fx1; x2;:::; xn−rg is a basis for nullspace(A). Remark Such an expression is called the general solution to the homogeneous linear system. Row Space, Col Space, Nonhomogeneous linear systems and Rank-Nullity Math 240 Row Space Now consider a nonhomogeneous linear system and Column Space Ax = b The Rank-Nullity Theorem where A be an m × n matrix and b is not necessarily 0. Homogeneous linear systems Nonhomogeneous Theorem linear systems I If b is not in colspace(A), then the system is inconsistent. I If b 2 colspace(A), then the system is consistent and has I a unique solution if and only if rank(A) = n. I an infinite number of solutions if and only if rank(A) < n. Geometrically, a nonhomogeneous solution set is just the corresponding homogeneous solution set that has been shifted away from the origin. Row Space, Col Space, Structure of a nonhomogeneous solution set and Rank-Nullity Math 240 Row Space Theorem and Column In the case where rank(A) = r < n and b 2 colspace(A), then Space The all solutions are of the form Rank-Nullity Theorem x = c1x1 + c2x2 + ··· + cn−rxn−r + xp; Homogeneous linear systems | {z } Nonhomogeneous = xc + xp linear systems where xp is any particular solution to Ax = b and fx1; x2;:::; xn−rg is a basis for nullspace(A). Remark The above expression is the general solution to a nonhomogeneous linear system. It has two components: I the complementary solution, xc, and I the particular solution, xp..