2.1 Linear Transformations 1

2.1 Linear Transformations 1 - 2.1 2 Linear Transformations...

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2.1. § 2. Linear Transformations and Matrices 2.1. Linear Transformations, Null Spaces, and Ranges 1. (a) T (b) F ( ) If x, y V and c F , T( x + y ) =T( x )+T( y ) and T( cx ) = c T( x ), then T is a linear transformation (c) F ( ) T is linear and one-to-one if and only if (d) T ( ) T(0 V ) =T(0 V + 0 V ) =T(0 V )+T(0 V ) T(0 V ) = 0 W (e) F ( ) p.70 Theorem 2.3 (f) F ( ) T is linear and one to one, then (g) T ( ) p.73 Corollary to Theorem 2.6 (h) F ( ) p.72 Theorem 2.6 2. (1) T (( a 1 , a 2 , a 3 ) + ( b 1 , b 2 , b 3 )) = T ( a 1 + b 1 , a 2 + b 2 , a 3 + b 3 ) = (( a 1 + b 1 ) - ( a 2 + b 2 ) , 2( a 3 + b 3 )) = ( a 1 - a 2 ) , 2 a 3 ) + ( b 1 - b 2 ) , 2 b 3 ) = T ( a 1 , a 2 , a 3 ) + T ( b 1 , b 2 , b 3 ) T ( c ( a 1 , a 2 , a 3 )) = T ( ca 1 , ca 2 , ca 3 ) = ( ca 1 - ca 2 , 2 ca 3 ) = c ( a 1 - a 2 , 2 a 3 ) = cT ( a 1 , a 2 , a 3 ) Thus, T is linear. (2) N(T)=span { (1 , 1 , 0) } R(T)=span { (1 , 0) , (0 , 1) } 55 PNU-MATH
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2.1. (3) Dim(V)=nullity(T) + rank(T)= 1 + 2 (4) T is not one-to-one ( N(T) 6 = { 0 } = { (0 , 0 , 0) } ) T is onto ( 2=rank(T)=dim(W)=2) 3. (1) T is linear. (2) N(T)=span { (0 , 0) } R(T)=span { (1 , 0 , 0) , (0 , 0 , 1) } (3) Dim(V)=nullity(T) + rank(T)= 0 + 2 (4) T is one-to-one ( N(T)= { 0 } ) T is not onto ( 2=rank(T) 6 = dim(W)=3) 4. (1) T is linear. (2) N(T)=span 1 0 0 0 , 0 1 0 0 , R(T)=span 1 0 0 0 0 0 , 0 1 0 0 0 0 , 0 0 1 0 0 0 , 0 0 0 1 1 1 (3) Dim(V)=nullity(T) + rank(T)= 4 + 2 (4) T is not one-to-one ( N(T) 6 = { 0 } ) T is not onto ( 2=rank(T) 6 = dim(W)=4) 5. (1) T is linear. 56 PNU-MATH
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2.1.
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