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**Unformatted text preview: **4. LINEAR TRANSFORMATIONS DR. TAEIL YI Linear Algebra with Applications 6th ed.-Steven J. Leon- LECTURE NOTE 1 2 DR. TAEIL YI 4.1 Definition and Examples Linear Transformation A mapping L from a vector space V into a vector space W is said to be a linear transformation or a linear operator if L ( v 1 + v 2 ) = L ( v 1 )+ L ( v 2 ) for all v 1 , v 2 V and for all scalars and . L is a linear operator if and only if L satisfies the following: (i) L ( v 1 + v 2 ) = L ( v 1 ) + L ( v 2 ) (ii) L ( v ) = L ( v ) A mapping L from a vector space V into a vector space W will be denoted L : V W . Ex 1. Let L be the operator defined by L ( x ) = 3 x for each x R 2 . Prove that L is a linear transformation on R 2 . In general, If is a positive scalar, the linear transformation F ( x ) = x can be thought of as a stretching or shrinking by a factor of . Ex 2. Consider the mapping L defined by L ( x ) = x 1 e 1 for each x R 2 . Prove that L is a linear transformation. Ex 3. Let L be the operator defined by L ( x ) = ( x 1 ,- x 2 ) T for each x = ( x 1 ,x 2 ) T in R 2 . Prove that L is a linear transformation. Ex 4. Consider the mapping L defined by L ( x ) = (- x 2 ,x 1 ) T for each x = ( x 1 ,x 2 ) T in R 2 . Prove that L is a linear transformation. Ex 5. Consider the mapping M defined by M ( x ) = ( x 2 1 + x 2 2 ) 1 / 2 . Prove that M is NOT a linear transformation. Ex 6. Prove that the mapping L : R 2 R 1 defined by L ( x ) = x 1 + x 2 is a linear transformation. Ex 7. Prove that the mapping L : R 2 R 3 defined by L ( x ) = ( x 2 ,x 1 ,x 1 + x 2 ) T is a linear transformation. If A is any m n matrix, we can define a linear operator L A from R n to R m by L A ( x ) = A x for each x R n . 4. LINEAR TRANSFORMATIONS 3 Ex. Prove that the operator L A is linear. (That is, each m n matrix A is a linear operator from R n to R m .) If L is a linear operator mapping a vector space...

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