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Lect20

# Lect20 - Chapter 3 Vector Spaces Math1111 Row Space Column...

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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 ) .

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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.

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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 .

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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 ) .

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

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