Solutions: 6.2 FE 1: If f is linear what is f( Solution: ) = FE 6: The fuction g : R 2 R 2 is linear Since = what is g( Solution:
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1 Solutions: 6 FE : If f is linear what is f(? f( 6 FE 6: The fuction g : R R is linear ( 5 ( g( g( Since ( 5 ( + ( what is g( (? 5g ( + g( ( ( + ( ( g( g(5 6 FE 9: Multiply: (a (b (c [ ] ( [ ] ( [ ] 6 7
2 (d 6 (a (b (c (d ( 7 ( 5 ( FE : Carry out the following matrix multiplications For each problem, say what the function represented by each matrix does to the stard basis vectors what the product of the two matrices would do to these vectors (a (b (c [ 5 ] [ ] (a [ ] 6 68
3 (b (c [ ] [ ] 5 9 To see what the matrices to to the stard basis vectors look at the columns of the matrices 6 FE : Multiply: (a (b (c (d [ ] (a (b (c (d [ ] 5 7 [ ]
4 6 FE : The function f : R R is linear ( ( ( (a If f( f( 5 ( (b What is f(? ( (c g : R R is also a linear function If g( matrix of g f? ( 5, find the matrix representing f 5 ( g( ( ( 7, what is the (a (b (c [ ] [ ] ( 7 ( 6 7 [ ] 5 6 FE : Black-lip oysters (Pinctada margaritifera are born male, but many become female later in life (a phenomenon known as protrous hermaphroditism We can therefore divide their population into three life stages: juveniles (which are all male, adult males, adult females Assume the following: - Each year, about 9% of juveniles remain juveniles, 9% grow to become adult males, % grow into adult females The rest die - Each year, about % of adult males become female, about % of them die - About % if adult females die each year Females never change back into males - Each female lays enough eggs to yield about juveniles per year Write a discrete time matrix model based on these assumptions J n+ 9 J n M n M n F n+ 9 F n
5 6: Consider the matrix M [ ] ( (a Compute Me, Me, M ( (b Draw e, e,, the vectors you obtained in the first part of this problem ( (c Describe what M does to e, e (d What will M do to other vectors that lie along the X axis? The Y axis? (e What will M do to the vectors that do not lie along the axes? (a [ ] ( ( [ ] ( ( [ ] ( ( (b See attached picture (c M doubles the length of e triples the lenght of e M stretches the X direction by a factor of in the Y direction ( by a factor of in (d M will double the length of vectors on the X axis M will triple the length of vectors on the Y axis (e M will stretch them by a factor of in the X direction by a factor of in the Y direction 6: One of the eigenvalues of the matrix M is, a corresponding eigenvector is V Find MV ( MV V ( ( 6 5
6 6: Compute the eigenvalues of the following matrices: [ ] 5 [ ] [ ] 5 : The characteristic equation is So the two eigenvalues are λ 7λ + λ (7 ± 676, 98 [ ] : The characteristic equation is λ 6λ + 5 So the two eigenvalues are λ, 5 65: Find the eigenvectors of the matrices whose eigenvalues you found in Exercise 6 [ ] 5 : The eigenvector for λ (7 +, we get that Let X, then So X + 5Y (7 + X X + 5Y (7 + Y Y + ( + is an eigenvector for λ (7 + For λ (7, we get that Let X, then So ( X + 5Y (7 X X + 5Y (7 Y Y is an eigenvector for λ (7 [ ] : The eigenvector for λ, we get that X + Y X X + Y Y 6
7 This gives that X Y So the eigenvector is any vector satisfying this equation So For λ 5, we get that This gives that X + Y 5X X + Y 5Y X Y So the eigenvector is any vector satisfying this equation So ( is an eigenvector for λ ( is an eigenvector for λ 5 6 FE : If M is a matrix is an eigenvector of M with corresponding eigenvalue 5, what is M? 5 M FE : If f : R R is a linear function is an eigenvalue of f with corresponding eigenvector v 7, what is f(v? f( FE : If A [ ] 7 has an eigenvector 8 8 ( What is its corresponding eigenvalue? [ ] ( So the eigenvalue is ( ( 6 7
8 6 FE 5: Which of the following are eigenvectors of eigenvalues? ( (a ( (b ( (c ( (d [ ] 7 5? What are their corresponding 8 (a (b (c (d This is not a multiple of This is a multiple The eigenvalue is Not a multiple so not an eigenvector This is a multiple The eigenvalue is [ ] ( ( ( so not an eigenvector [ ] ( [ ] ( [ ] ( ( 6 ( 6 ( 6 FE 6: Compute the eigenvalues, if they exist, eigenvectors of the following matrices: [ ] 7 9 (a [ ] (b 6 8
9 (c (d (e (f [ ] 5 5 [ ] [ ] 5 9 [ ] ( (a The eigenvalues are λ, The eigenvectors are (b The eigenvalues are λ ± ( 7 The eigenvectors are (c No eigenvalues (d The eigenvalues are λ ± The eigenvectors are (e The eigenvalues are λ ± 5 The eigenvectors are (f The eigenvalues are λ, The eigenvectors are ( ( + 7 ( + ( 5 5 ( ( ( ( ( 5+ 5 ( 7 9
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