MA 0540 fall 2013, Row operations on matrices December 2, 2013 This is all about m by n matrices (of real or complex numbers). If A is such a matrix then A n m corresponds to a linear map F ! F , which we will denote by TA. The map is defined by matrix n multiplication: TA(X) = AX, where the vector X 2 F is thought of as an n by 1 matrix and the m m by 1 matrix AX is thought of as a vector in F . n At times I will view the matrix as a list of m vectors R1;:::;Rm, all in F (the rows). At other m times I will view it as a list of n vectors C1;:::;Cn, all in F (the columns). 1 The rank of a matrix 1.1 The column space of a matrix m The subspace of F spanned by the columns of A is called the column space of A, and the rank of this subspace is called the column rank of A. The column space is the same as the range of TA, because TA takes the vector (x1; : : : ; xn) to x1C1 + ::: + xnCn. Thus the column rank of the matrix A is the dimension of the range of the linear map TA. Of course, the column rank is no bigger than m, and it is equal to m if and only if TA is surjective. 1.2 The row space of a matrix n The subspace of F spanned by the rows of A is called the row space of A, and its rank is called the row rank of A. 1 1.3 Equality of row rank and column rank In fact we know that the dimension of the column space of A is always the same as the dimension m n of the row space. We proved this before by using the nullspace of the linear map F ! F that corresponds to the transpose matrix. We will see a different proof below. 2 Row operations How can we determine the rank of a matrix A? How can we find a basis for the nullspace of TA? Row operations give answers to these and other questions. 2.1 The operations For this discussion, think of an m by n matrix as a list of its rows Ri. There are three types of things we can do to a matrix that are called row operations. 1. Interchange Ri with Rj for some i and j 6= i. 2. For some i, replace Ri by cRi for some scalar c 6= 0. Leave the other rows unchanged. 3. For some i and j 6= i, replace Ri by Ri + cRj for some scalar c. Leave the rows other than Ri unchanged. We say that two m by n matrices A and B are row-equivalent if it is possible to change A into B by a sequence of row operations. It is clear that row operations can be reversed: if some row operation turns A into B then some row 0 operation turns B into A. For example, if you get B be replacing the ith row Ri by Ri = Ri + cRj 0 0 then you can recover A from B by replacing the new ith row Ri by Ri + (−c)Rj = Ri. Here is one way to look at this: Any row operation applies a certain linear operator to every column of the matrix: if the columns of the matrix A are C1;:::;Cn then the columns of the new matrix are PC1;:::;PCn, where P is a certain m by m matrix. (The new matrix is PA.) For any row operation of any of the three types, there is some m by m matrix P such that this is true. The matrix P is always invertible. 2.2 The effect of row operations on the row space and column space of A If A and B are row-equivalent, then they have the same row space. In fact, when a matrix is altered by a row operation then every new row belongs to the span of the old rows, so the new row space 2 is contained in the old row space; and since row operations can be reversed it is also true that the old row space is contained in the new row space. Therefore the new row space and the old row space are the same. It follows, of course, that the row rank of a matrix does not change when we perform row operations on it. Row operations do change the column space of a matrix, but nevertheless they do not change its dimension. To see this, suppose that that a row operation corresponds to the invertible matrix P as above. The row space of PA has the same dimension as the row space of A because the one m subspace of R is obtained from the other by applying an invertible operator TP to it. So, just like the row rank, the column rank of A is unaltered by row operations. 3 Echelon matrices We call an m by n matrix A an echelon matrix if it has the following form: For some number r (an integer satisfying 0 ≤ r ≤ m) there are numbers c(1); : : : ; c(r) satisfying 1 ≤ c(1) < : : : < c(r) ≤ n such that: When i > r then ai;j = 0 for all j. For every i from 1 to r we have ai;c(i) = 1. For every i from 1 to r and every j < c(i) we have ai;j = 0. For every i from 1 to r and every k < i we have ak;c(i) = 0. In other words, Ri = 0 if i > r; each row after the first r rows is entirely zero. If i ≤ r then in the row Ri the first nonzero entry is a 1, and it occurs in column number c(i), where c(i) is an increasing function of i. In the column Cc(i) everything above the 1 in row i is a zero. Note that inside this matrix there is an r by r submatrix that is an identity matrix; it is in the rows numbered 1; 2; : : : ; r and the columns numbered c(1); c(2); : : : ; c(r). 3 3.1 Rank of an echelon matrix We now show directly that for an echelon matrix both the row rank and the column rank are equal to the number called r above, the number of rows that are not entirely made of zeroes. To see that the row rank is r, it suffices to show that the first r rows are linearly independent. For this, suppose we have a linear relation x1R1 + ::: + xrRr = 0, i.e. a linear relation x1a1;j + x2a2;j + ::: + xrar;j = 0 valid for every j from 1 to n. For any i from 1 to r we may see that xi = 0 by taking j to be c(i): this yields the equation x1a1;c(i) + x2a2;c(i) + ::: + xrar;c(i) = 0, which tells us that xi = 0, since the only one of a1;c(i); : : : ; ar;c(i) that is not zero is ai;c(i) = 1. To see that the column rank of an echelon matrix is also r we can observe that the column space m is the r-dimensional subspace of F consisting of all vectors (x1; : : : ; xm) such that for every i > r the number xi is zero. It is contained in that subspace because each column Cj is that subspace (i.e. the numbers aij are all zero for i > r), and it is all of that subspace because the columns Cc(1);:::;Cc(r) are the obvious basis of that subspace. 3.2 Every matrix is row-equivalent to some echelon matrix Here is a procedure (\row reduction of a matrix") for doing row operations on a matrix to put it in echelon form. Look at the first column. If it is entirely zero, then ignore it and go to work on the remaining m by n − 1 matrix. If some row operations turn that submatrix into an echelon matrix, say E, then these same operations will also turn A into an echelon matrix (E with one extra row of zeroes on the left). If the first column is not entirely zero, then arrange for the upper left entry a1;1 to be not 0, by interchanging the first row with some other row if necessary. Now that a 6= 0, multiply the first row by 1 . So in the new matrix a = 1. 1;1 a1;1 1;1 Now for each i 6= 1 do a row operation to make ai;1 into 0. (Replace Ri by Ri − ai;1R1.) At this point the first column has 1 at the top and all zeroes below. If m = 1 then we are done: this 1 by n matrix is in echelon form. Also if n = 1 then we are done: this m by 1 matrix is jun echelon form. So assume m > 1 and n > 1. Temporarily ignore the first column and the first row and look at the remaining m − 1 by n − 1 matrix. Do row operations on it until it is in echelon form. These same operations performed on the original matrix will not change the first row, and they will not affect the first column either (because ai;1 = 0 for i > 1). 4 At this point our m by n matrix is almost in echelon form.