Lect18 - Chapter 3. Vector Spaces Math1111 Row Space &...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Row Space & Column space Definition Definition Let A be an m × n matrix. Each row a ( i ) : = a ( i ,: ) , i = 1, ··· , m , is a vector in R 1 × n . We define the row space of A to be Span ( a ( 1 ) , ··· , a ( m )) and denoted by r ( A ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Definition Definition Let A be an m × n matrix. Each row a ( i ) : = a ( i ,: ) , i = 1, ··· , m , is a vector in R 1 × n . We define the row space of A to be Span ( a ( 1 ) , ··· , a ( m )) and denoted by r ( A ) . Each column a j , j = 1, ··· , n , is a vector in R m . We define the column space of A to be Span ( a 1 , ··· , a n ) and denoted by c ( A ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Definition Definition Let A be an m × n matrix. Each row a ( i ) : = a ( i ,: ) , i = 1, ··· , m , is a vector in R 1 × n . We define the row space of A to be Span ( a ( 1 ) , ··· , a ( m )) and denoted by r ( A ) . Each column a j , j = 1, ··· , n , is a vector in R m . We define the column space of A to be Span ( a 1 , ··· , a n ) and denoted by c ( A ) . Remark r ( A ) is a subspace of R 1 × n , c ( A ) is a subspace of R m . Chapter 3. Vector Spaces Math1111 Row Space & Column space Example 1 Example . Let U =       1 2 3 1 1 1 2 2 1 1       . Let u ( i ) be the i th row of U and u j be its j th column. Show that (i) r ( U ) = Span ( u ( 1 ) , u ( 2 ) , u ( 3 )) , (ii) c ( U ) = Span ( u 1 , u 2 , u 4 ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Example 1 Example . Let U =       1 2 3 1 1 1 2 2 1 1       . Let u ( i ) be the i th row of U and u j be its j th column. Show that (i) r ( U ) = Span ( u ( 1 ) , u ( 2 ) , u ( 3 )) , (ii) c ( U ) = Span ( u 1 , u 2 , u 4 ) . Proof. (i) By definition, r ( U ) = Span ( u ( 1 ) , u ( 2 ) , u ( 3 ) , ) . Clearly, Span ( u ( 1 ) , u ( 2 ) , u ( 3 ) , ) = Span ( u ( 1 ) , u ( 2 ) , u ( 3 )) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Example 1 Example . Let U =       1 2 3 1 1 1 2 2 1 1       ....
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Lect18 - Chapter 3. Vector Spaces Math1111 Row Space &...

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