Lect20 - Chapter 3. Vector Spaces Math1111 Row Space &...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Row Space & Column space Rank Definition Let A be a matrix. The rank of A , denoted by rank ( A ) , is defined by rank ( A ) = dim r ( A ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Rank Definition Let A be a matrix. The rank of A , denoted by rank ( A ) , is defined by rank ( A ) = dim r ( A ) . Remark. In many books, dim r ( A ) is called the row rank of A , and dim c ( A ) is called the column rank of A . Chapter 3. Vector Spaces Math1111 Row Space & Column space Rank Definition Let A be a matrix. The rank of A , denoted by rank ( A ) , is defined by rank ( A ) = dim r ( A ) . Remark. In many books, dim r ( A ) is called the row rank of A , and dim c ( A ) is called the column rank of A . Theorem 3.6.4 says that row rank = column rank. The rank of A is defined to be their common value. Chapter 3. Vector Spaces Math1111 Row Space & Column space Rank Definition Let A be a matrix. The rank of A , denoted by rank ( A ) , is defined by rank ( A ) = dim r ( A ) . Remark. In many books, dim r ( A ) is called the row rank of A , and dim c ( A ) is called the column rank of A . Theorem 3.6.4 says that row rank = column rank. The rank of A is defined to be their common value. Definition Let A be a matrix. The nullity of A is the dimension of the nullspace of A . Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorem 3.6.5 Theorem 3.6.5 (The Rank-Nullity Theorem) Let A be an m × n matrix. Then rank ( A ) + dim N ( A ) = n . Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorem 3.6.5 Theorem 3.6.5 (The Rank-Nullity Theorem) Let A be an m × n matrix. Then rank ( A ) + dim N ( A ) = n . Proof. Let U be the reduced row echelon form of A . Then U = EA and Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorem 3.6.5 Theorem 3.6.5 (The Rank-Nullity Theorem) Let A be an m × n matrix. Then rank ( A ) + dim N ( A ) = n . Proof. Let U be the reduced row echelon form of A . Then U = EA and so (i) dim N ( A ) = dim N ( U ) , and (ii) rank ( A ) = rank ( U ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorem 3.6.5 Theorem 3.6.5 (The Rank-Nullity Theorem) Let A be an m × n matrix. Then rank ( A ) + dim N ( A ) = n ....
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Lect20 - Chapter 3. Vector Spaces Math1111 Row Space &...

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