2 PART I: Theoretical problems which require detailed solu- tions 0 2 0 Step 1. Create your Personal matrix. Set id := the last digit of your ID number. Change the matrix A. given below to your person- alized matrix A assigned for you, by replacing "id" with the last digid of your ID number (x - id) 0 Ao = 0 id 0 id (2-7) 0 0 For example, if your ID number is ******7, then your personalized matrix is A = 27 72 Make sure you use the last digit of your personal ID, do not work with the personalized matrix of your friend! 1. Problem. (18 points) Let A be the personalized matrix, you have written above. (a) Compute the determinant, det A. Explain with details the method and the steps you perform to compute the determinant (depending the method you use). (b) Present the determinant det A as a polynomial f(x) of degree 3, find all roots of f(x) (with their multiplicity) and present f(x) as a product of linear factors f(x) = (1 - 11)(x - 22)(x - 13). Remark. A direct answer of the form det A = (x - 1)(x - 1)(x + 2) without showing the steps in your computation will not receive credit (the credit will be 0). 2. Problem. (18 points) Let A be your personalized matrix you have written above. Study how the null space of A depends on x. Find all values of such that (a) Nul A is the origin only; (b) Nul A is a plane through the origin; find a basis of this plane and an equation of the plane; (c) Nul A is a line through the origin; find a vector that spans this line. Every answer should be motivated. Prevention: an answer in the spirit "Nul A is a plane (resp. a line) if and only if x = 1" but without explanation why is this a plane (resp. a line) will not receive any credit. Remark. For the solution of Problem 2 you may use either your results from Problem 1, if it is applicable, or you may use a different independent method. 3. Problem. (19 points) Let A be your personalized matrix, found above. Let {aj, a2, az} be the columns of A. Let V be the column space of A, V = ColA (1) Find all values of r such that the set {ai, a2, a3} of columns of A is linearly independent. Motivate your ans (2) Determine for which values of 2 i. there is an equality of spaces V = R3 ii. V is a plane through the origin; iii. V is a line through the origin; V is the origin only. In each of the cases (a), (b), (c) find a basis of V, consisting of some of the vectors aj, aj, a3. Remark. For the solution of Problem 3 you may use either your results from Problem 1, if it is applicable, or you may chose a different approach. Every answer must be motivated. Make sure your answers agree with the Fundamental Theorem of Linear Algebra Part 1, (rank and nullity theorem) and check whether your answe for Problem 3 are compatible with the results in Problem 2. Prevention. An answer in the spirit "V is a plane (resp. a line) if and only if I = a concrete number" but without explanation why is this a plane (resp. a line) will not receive a credit. swers 2 PART I: Theoretical problems which require detailed solu- tions 0 2 0 Step 1. Create your Personal matrix. Set id := the last digit of your ID number. Change the matrix A. given below to your person- alized matrix A assigned for you, by replacing "id" with the last digid of your ID number (x - id) 0 Ao = 0 id 0 id (2-7) 0 0 For example, if your ID number is ******7, then your personalized matrix is A = 27 72 Make sure you use the last digit of your personal ID, do not work with the personalized matrix of your friend! 1. Problem. (18 points) Let A be the personalized matrix, you have written above. (a) Compute the determinant, det A. Explain with details the method and the steps you perform to compute the determinant (depending the method you use). (b) Present the determinant det A as a polynomial f(x) of degree 3, find all roots of f(x) (with their multiplicity) and present f(x) as a product of linear factors f(x) = (1 - 11)(x - 22)(x - 13). Remark. A direct answer of the form det A = (x - 1)(x - 1)(x + 2) without showing the steps in your computation will not receive credit (the credit will be 0). 2. Problem. (18 points) Let A be your personalized matrix you have written above. Study how the null space of A depends on x. Find all values of such that (a) Nul A is the origin only; (b) Nul A is a plane through the origin; find a basis of this plane and an equation of the plane; (c) Nul A is a line through the origin; find a vector that spans this line. Every answer should be motivated. Prevention: an answer in the spirit "Nul A is a plane (resp. a line) if and only if x = 1" but without explanation why is this a plane (resp. a line) will not receive any credit. Remark. For the solution of Problem 2 you may use either your results from Problem 1, if it is applicable, or you may use a different independent method. 3. Problem. (19 points) Let A be your personalized matrix, found above. Let {aj, a2, az} be the columns of A. Let V be the column space of A, V = ColA (1) Find all values of r such that the set {ai, a2, a3} of columns of A is linearly independent. Motivate your ans (2) Determine for which values of 2 i. there is an equality of spaces V = R3 ii. V is a plane through the origin; iii. V is a line through the origin; V is the origin only. In each of the cases (a), (b), (c) find a basis of V, consisting of some of the vectors aj, aj, a3. Remark. For the solution of Problem 3 you may use either your results from Problem 1, if it is applicable, or you may chose a different approach. Every answer must be motivated. Make sure your answers agree with the Fundamental Theorem of Linear Algebra Part 1, (rank and nullity theorem) and check whether your answe for Problem 3 are compatible with the results in Problem 2. Prevention. An answer in the spirit "V is a plane (resp. a line) if and only if I = a concrete number" but without explanation why is this a plane (resp. a line) will not receive a credit. swers