c2s2 - H~x = ~ 0 as before. Example. Page 140 number 4....

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2.2 The Rank of a Matrix 1 Chapter 2. Dimension, Rank, and Linear Transformations 2.2. The Rank of a Matrix Note. In this section, we consider the relationship between the dimen- sions of the column space, row space and nullspace of a matrix A . Theorem 2.4. Row Rank Equals Column Rank. Let A be an m × n matrix. The dimension of the row space of A equals the dimension of the column space of A . The common dimension is the rank of A . Note. The dimension of the column space is the number of pivots of A when in row-echelon form, so by page 129, the rank of A is the number of pivots of A when in row-echelon form.
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2.2 The Rank of a Matrix 2 Note. Finding Bases for Spaces Associated with a Matrix. Let A be an m × n matrix with row-echelon form H . (1) for a basis of the row space of A , use the nonzero rows of H (or A ), (2) for a basis of the column space of A , use the columns of A correspond- ingtotheco lumnso f H which contain pivots, and (3) for a basis of the nullspace of A use
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Unformatted text preview: H~x = ~ 0 as before. Example. Page 140 number 4. Theorem 2.5. Rank Equation. Let A be m n with row-echelon form H . (1) The dimension of the nullspace of A is nullity( A ) = (# free variables in solution of A~x = ~ 0) = (# pivot-free columns of H ) . (2) rank( A ) = (# of pivots in H ) . (3) Rank Equation: rank( A ) + nullity( A ) = # of columns of A. 2.2 The Rank of a Matrix 3 Theorem 2.6. An Invertibility Criterion. An n n matrix A is invertible if and only if rank( A ) = n. Example. Page 141 number 12. If A is square, then nullity( A ) = nullity( A T ). Proof. The column space of A is the same as the row space of A T , so rank( A ) = rank( A T ) and since the number of columns of A equals the number of columns of A T , then by the Rank Equation: rank( A ) + nullity( A ) = rank( A T ) + nullity( A T ) and the result follows. QED...
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c2s2 - H~x = ~ 0 as before. Example. Page 140 number 4....

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