This proves that row operations do not change the row space of a matrix. In other words if we look at the row vectors of A can be written as a linear combination of It turns out that row operations do not change We find a basis for the row space of A from rref(A)? In order toĪnswer this question, we must understand how row operations change the Row1 of A = and row2 of A = Method 2 for finding a basis for the row space of A: Weįound a basis for the column space of A by computing rref(A). Thus if we goīack to A we now know that row 3 of A is the sum of rows 1 and 2 and rowĤ of A is 2*row1 + 3*row2 and thus we know that the row space of A hasĭimension 2 as the theorem predicts. So we see that column 3 of A T is the sum of columns 1Īnd 2 and column 4 is the 2*column 1 + 3*column 2. We could take these row vectors and put them into a matrix as Get e A (3,2) e 10 e is the element in the 3,2 position (third row, second column) of A. We need to understand any linear relations that hold among the rows ofĪ. Method 1 for finding a basis for the row space of A: To find a basis for the row space of A we could proceed in two ways: Since column 1 and column 2 of rref(A) are clearly independent, the same We see that column 3 = column 2 - column 1 and column 4 = column 1 + columnĢ, we conclude that the same relations hold among the columns of A. Of rref(A) because A and rref(A) have the same kernel. The same relations hold among the columns of A as hold among the columns This is the same asįinding the kernel of A and we do this by bringing A into reduced form: Linear relations that hold among the columns. In order to understand the column space of A we need to understand any That is, the rank of A tells us the dimensionĮxample: Find a basis for the row space and for the column Theorem 2: The rank of A is equal to the number of linearly Since the columns of the transpose ofĪ are the same as the rows of A, our theorem 1 is equivalent to The matrix A has and therefore is equal to the dimension of the image of Later we learned that this tells us how many linearly independent columns We first defined the rank of A to be the number of leading 1's in rref(A). Theorem 1: The rank of A is equal to the rank of A T
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