ila0306 - 184 Chapter 3 Vector Spaces and Subspaces 3.6...

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184 Chapter 3. Vector Spaces and Subspaces 3.6 Dimensions of the Four Subspaces The main theorem in this chapter connects rank and dimension . The rank of a matrix is the number of pivots. The dimension of a subspace is the number of vectors in a basis. We count pivots or we count basis vectors. The rank of A reveals the dimensions of all four fundamental subspaces. Here are the subspaces, including the new one. Two subspaces come directly from A , and the other two from A T : Four Fundamental Subspaces 1. The row space is C .A T / , a subspace of R n . 2. The column space is C .A/ , a subspace of R m . 3. The nullspace is N .A/ , a subspace of R n . 4. The left nullspace is N .A T / , a subspace of R m . This is our new space. In this book the column space and nullspace came first. We know C .A/ and N .A/ pretty well. Now the other two subspaces come forward. The row space contains all combinations of the rows. This is the column space of A T . For the left nullspace we solve A T y D 0 —that system is n by m . This is the nullspace of A T . The vectors y go on the left side of A when the equation is written as y T A D 0 T . The matrices A and A T are usually different. So are their column spaces and their nullspaces. But those spaces are connected in an absolutely beautiful way. Part 1 of the Fundamental Theorem finds the dimensions of the four subspaces. One fact stands out: The row space and column space have the same dimension r (the rank of the matrix). The other important fact involves the two nullspaces: N .A/ and N .A T / have dimensions n r and m r , to make up the full n and m . Part 2 of the Fundamental Theorem will describe how the four subspaces fit together (two in R n and two in R m / . That completes the “right way” to understand every A x D b . Stay with it—you are doing real mathematics. The Four Subspaces for R Suppose A is reduced to its row echelon form R . For that special form, the four subspaces are easy to identify. We will find a basis for each subspace and check its dimension. Then we watch how the subspaces change (two of them don’t change!) as we look back at A . The main point is that the four dimensions are the same for A and R . As a specific 3 by 5 example, look at the four subspaces for the echelon matrix R : m D 3 n D 5 r D 2 2 4 1 3 5 0 7 0 0 0 1 2 0 0 0 0 0 3 5 pivot rows 1 and 2 pivot columns 1 and 4 The rank of this matrix R is r D 2 ( two pivots ). Take the four subspaces in order.
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3.6. Dimensions of the Four Subspaces 185 1. The row space of R has dimension 2 , matching the rank. Reason: The first two rows are a basis. The row space contains combinations of all three rows, but the third row (the zero row) adds nothing new. So rows 1 and 2 span the row space C .R T / . The pivot rows 1 and 2 are independent. That is obvious for this example, and it is always true. If we look only at the pivot columns, we see the r by r identity matrix.
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