Rank. Rank. Definition. The set of all linear combination of the row vectors of a matrix A is called the row space of A and is denoted by Row A

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1 Rank Rank he rank of a matrix is the maximum number of independent rows (or the maximum number of independent columns) Definition he set of all linear combination of the row vectors of a matrix A is called the row space of A and is denoted by Row A ( ) Note we also refer to Row A as Col A and R A

2 6 Example A = 6 6 [ ] [ 6 ] [ 6] r = 6 r = r = A = Instead of working with row vectors we usually work with the column vectors of the transposed matrix Remarks Row A = Col A When we use row operations to reduce matrix A to matrix B we are taking linear combinations of the rows of A to come up with B. We could reverse this process and use row operations on B to be back to A. Because of this the row space of A equals to row space of B. heorem If two matrices A and B are row equivalent then their row spaces are the same If B is in echelon form the nonzero rows of B form a basis for the row space of A as well as B 6

3 Example A = Col A = Row A = 6 dim Col A = dim Row A = 7 Example 6 x 6 x = x + x x x 6 A = 6 6 x = x + 6x x = x x free dim Nul A = 6 Nul A = 8 Example In our example we saw: 6 A = 6 6 dim Col A = number of pivots of A = number of nonzero rows of B = dim Row A dim Nul A = number of free variables = number of nonpivot columns of A 9

4 Definition he rank of A is the dimension of the column space rank A = dim Col A = dim Row A ( ) ( ) rank A = dim R A = dim R A Remarks In our previous example we saw ( rank A) + ( dim Nul A) = ( n) + = rank A + dim Nul A = n # of pivot # of nonpivot # of columns + columns = columns of A of A of A heorem Rank hm he dimensions of the column space and the row space of an m n matrix Aare equal. his common dimension satisfies the equation rank A + dim Nul A = n and since Row A = Col A rank A= rank A

5 Example Suppose a 8 matrix A has rank Find dim Nul A dim Row A and rank A using rank A + dim Nul A = n dim Nul A = n rank A= 8 = dim Row A = rank A = rank A = rank A = Example For a 9 matrix A find the smallest possible value of dim Nul A using rank A + dim Nul A = n we have dim Nul A = n rank A for the smallest null space we must have the largest column space i.e. the largest rank 9 largest rank = 9 using rank A = 9 n= smallest dim Nul A = 9 = Example 6 6 Col A = x = x Nul A = x = x x free 6 Row A =

6 Example A = 6 so another basis for = Row A 6 R( A ) 6 Compare and 6 = + = Example 6 A = 6 So for A we have n = and rank = but rank A +dim Nul A = n hence dim Nul A = ( ) N A = We observe that A: R R 7 Example In R N( A) = 6 = With dim R = and dim( line) = We are left with a plane dim = ( ) 6 is a line in R 8 6

7 Example In R 6 R( A) = wo vectors in R form a plane 9 Example 6 We observe A : R R Col A = RowA = 6 In R two vectors form a plane. Hence A maps everything in R into a plane in R Practice A scientist solves a homogeneous system of equations in variables and finds that exactly of the unknowns are free variables. Can the scientist be certain that any nonhomogeneous system with the same coefficients has a solution? With n= dim Nul A= we know that dim Col A is (because of the Rank hm) dim Col A is there is a pivot in each row there is a solution to any A x=bsystem. 7

8 Remarks A square matrix A n n is non-singular only if its rank is equal to n heorem 8 Invertible Matrix heorem Let A be a square n n matrix. he following statements are equivalent. i.e. for a given matrix A they are all either true or false We will add to this list but first we review the current list heorem 8 - Invertible Matrix heorem Let A be a square n n matrix. he following statements are equivalent. i.e. for a given matrix A they are all either true or false A is an invertible matrix A is row equivalent to I n A has n pivot positions he equation Ax = has only the trivial solution he columns of A are linearly independent he linear transformation x Ax is one-to-one he equation Ax = b is consistent for each b in R n he columns of A span R n he linear transformation x Ax maps R n onto R n here is a n x n matrix C such that AC = I n here is a n x n matrix D such that DA = I n A is an invertible matrix 8

9 heorem 8 - Invertible Matrix heorem Let A be a square n n matrix. he following statements are equivalent. i.e. for a given matrix A they are all either true or false Now we expand our list he columns of A form a basis for R n Col A = R n Dim Col A = n Rank A = n Null A = {} Dim Nul A = We refer to this theorem as the IM 9