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Lecture 29

MATH 125 Lecture 29: Lecture 29

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Department
Mathematics
Course
MATH 125
Professor
Scott
Semester
Fall

Description
Recap Let T: Rn Rp be a linear transformation. The standard matrix of T is the p x n matrix For any UE Rn, we have If S: Rp +Rm is another linear transformation, then the composition So T: R" RIT is defined by setting This is again a linear transformation. Solution Theorem 6.12: Let T: Rn Rp and S Rp Rm be linear transformations. Then the standard matrix of the com position SoT is equal to the product of the standard matrices of the individual transformations: Proof Let v e R n. Then Remark: More generally, if Ti, T2, TR are linear trans- formations for which the composition Tk o Tr o To o Th is defined, then o T2 o T T. T Example 6.13: Let T: R2 R4 and S: R4 Ra be the linear transformations given by T2 and S 4r T4 Recap Let T: Rn Rp be a linear transformation. The standard matrix of T is the p x n matrix For any UE Rn, we have If S: Rp +Rm is another linear transformation, then the composition So T: R" RIT is defined by setting This is again a linear transformation. Solution Theorem 6.12: Let T: Rn Rp and S Rp Rm be linear transformations. Then the standard matrix of the com position SoT is equal to the product of the standard matrices of the individual transformations: Proof Let v e R n. Then Remark: More generally, if Ti, T2, TR are linear trans- formations for which the composition Tk o Tr o To o Th is defined, then o T2 o T T. T Example 6.13: Let T: R2 R4 and S: R4 Ra be the linear transformations given by T2 and S 4r T4Find the standard matrix IS o T of the composition S o T using (1) the definition, and (2) matrix multiplication. Solution (1) Find (s oT)CET) an (s. T) EL') (2) In CT1 and [S] 1 Fina O 3 1 4 1 1 1 0 1 4 0 1 -A o 1 -1 O 2 Example 6.14: Find the standard matrix of the linear oper- ator T: R2 IR2 that performs a counter-clockwise rotation about the origin through radians followed by reflection about the y-axis. Solution: (TT/2) -sin( /2) otation T1) 2) cos /2 Reflection (T2) Find the standard matrix IS o T of the composition S o T using (1) the definition, and (2) matrix multiplication. Solution (1) Find (s oT)CET) an (s. T) EL') (2) In CT1 and [S] 1 Fina O 3 1 4 1 1 1 0 1 4 0 1 -A o 1 -1 O 2 Example 6.14: Find the standard matrix of the linear oper- ator T: R2 IR2 that performs a counter-clockwise rotation about the origin through radians followed by reflection about the y-axis. Solution: (TT/2) -sin( /2) otation T1) 2) cos /2 Reflection (T2)n about tu line y x reflecti Invertibilility of linear operators Definition: The linear operator In: Rn Rn defined by setting In(i) for all i E R" is called the identity map on R Since In e for all 1 S is n, the standard matrix of L is the n x n identity matrix In. ([In 3In) Definition: Let T: R" Rn be a linear operator on Rn We say that T is invertible if there exists another linear operator S: R" R" such that S o T and T o S I In this case we say that S is an inverse of T Note: As with square matrices, if T is invertible, then it has a unique inverse, and we denote it by T Example 6.15: The linear operator T: R2 R2 that re- ffects vectors about a line l through the origin is invertible. In fact, T T Cra Plechng twice is the identit Example 6.16: The linear operator T: R2 R2 given by counter-clockwise rotation about the origin through 0 radians is invertible. The inverse operator T 1 is given by clock Wisc. tation about the origin through radians The concept of invertibility for linear operators on R" is in fact equivalent to the concept of invertibility for n x n matrices: Theorem 6.17: A linear operator T: Rn R" is invertible if and only if its standard matrix Tj s invertible. Moreover, in this case, we have Example 6.18: The standard matrix of the linear operator T on R2 given by reflection about the z-axis is (Example 6.8) Since T 1 T (Example 6.15), we have T n about tu line y x reflecti Invertibilility of linear operators Definition: The linear operator In: Rn Rn defined by setting In(i) for all i E R" is called the identity map on R Since In e for all 1 S is n, the standard matrix of L is the n x n identity matrix In. ([In 3In) Definition: Let T: R" Rn be a linear operator on Rn We say that T is invertible if the
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