Sec5p4-W2022-SV

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Carleton University **We aren't endorsed by this school
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MATH 2107
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Mathematics
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Oct 17, 2023
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Ay¸ se Alaca MATH 2107 (Eigenvectors & Linear Transformations) 1 LINEAR ALGEBRA II LECTURE NOTES c Ay¸ se Alaca Section 5.4 EIGENVECTORS and LINEAR TRANSFORMATIONS Textbook: Linear Algebra and its Applications, 6E, By David C. Lay, Steven R. Lay, Judi J. McDonald Please do not repost these notes elsewhere. These lecture notes may be incomplete or contain errors/typos.
Ay¸ se Alaca MATH 2107 (Eigenvectors & Linear Transformations) 2 Eigenvectors of Linear Transformations Let V be a vector space. We define eigenvalues and eigenvectors of any linear trans- formation T : V 7-→ V . Definition: Let V be a vector space and T : V 7-→ V be a linear transformation. An eigenvector of T is a nonzero vector v V such that T ( v ) = λ v for some scalar λ . A scalar λ is called an eigenvalue of T if there is a nontrivial solution v of T ( v ) = λv ; such a vector v is called an eigenvector of T corresponding to λ . Example 1: Consider the linear transformation T : P 2 7-→ P 2 defined by T ( a + bx + cx 2 ) = a + ( a + b + c ) x + ( a + 4 b + c ) x 2 . Then, we have T ( x - 2 x 2 ) = - x + 2 x 2 = - ( x - 2 x 2 ) = λ 1 = - 1 T ( - 4 + x + 4 x 2 ) = - 4 + x + 4 x 2 = 1( - 4 + x + 4 x 2 ) = λ 2 = 1 T ( x + 2 x 2 ) = 3 x + 6 x 2 = 3( x + 2 x 2 ) = λ 3 = 3 So, x - 2 x 2 , - 4 + x + 4 x 2 and x + 2 x 2 are eigenvectors of T with corresponding eigenvalues - 1 , 1 , 3, respectively. In Section 4.2, we learned the matrix of a linear transformation T : R n 7-→ R m with respect to the standard bases of R n and R m respectively. In this section, we will learn more about the matrices of linear transformations, and then see the connections between its matrices, eigenvalues and eigenvectors.
Ay¸ se Alaca MATH 2107 (Eigenvectors & Linear Transformations) 3 The Matrix of a Linear Transformation We willl look at the matrix factorization A = PDP - 1 in terms of linear transforma- tions. Let V be an n -dimensional vector space, W be an m -dimensional vector space, B be a basis for V , and C be a basis for W . Let T : V W be a linear transformation. We know that for any v V , [ v ] B R n and [ T ( v )] C R m . Question: What is the connection between [ v ] B and [ T ( v )] C ? Answer: Let B = { b 1 , b 2 , . . . , b n } . Let v V . Then there exists scalars r 1 , r 2 , . . . , r n such that v = r 1 b 1 + r 2 b 2 + · · · + r n b n = [ v ] B = r 1 r 2 . . . r n . T ( v ) = T ( r 1 b 1 + r 2 b 2 + · · · + r n b n ) = r 1 T ( b 1 ) + r 2 T ( b 2 ) + · · · + r n T ( b n ) [ T ( v )] C = [ r 1 T ( b 1 ) + r 2 T ( b 2 ) + · · · + r n T ( b n )] C = r 1 [ T ( b 1 )] C + r 2 [ T ( b 2 )] C + · · · + r n [ T ( b n )] C . Since [ T ( b i )] C R m (1 i n ), we can write the above vector equation as a matrix equation [ T ( v )] C = h [ T ( b 1 )] C [ T ( b 2 )] C . . . [ T ( b n )] C i | {z } [ T ] C←-B r 1 r 2 . . . r n = [ T ] C←-B [ v ] B (1) The matrix [ T ] C←-B is a matrix representation of T , and it is called the matrix for T relative to the bases B and C .
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