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Problem 2 (10 pt) Consider real functions with a general form of: f(x) = 20 +...
2) The set S of all real-valued functions f(x) of a single real variable z is...
2) The set S of all real-valued functions f(x) of a single real variable z is a vector space. (a) Show that the set L of all real-valued linear functions f(x) = mx + b of a single variable x is a subspace of S. (b) Show tha (f(x), g(x))= | f(z)g(x)dx is an inner product on L. (c) Find an orthonormal basis for C with respect to the inner product defined in (b)
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ]...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
Question 1. Consider these real-valued functions of two variables: T In (x2 + y2) (a) () What is ...
Question 1. Consider these real-valued functions of two variables: T In (x2 + y2) (a) () What is the maximal domain, D, for the functions f and g? Write D in set notation. (ii) What is the range of f and g? Is either function onto? ii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z00, 2, 04 (Note: Use set notation, and draw a single contour diagram.) (v) Without finding...
Question 1. Consider these real-valued functions of two variables f(x, y) (a) (i) What is the max...
Question 1. Consider these real-valued functions of two variables f(x, y) (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? (iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: 20-0, 20-2, 20-4 (Note: Use set notation, and draw a single contour diagram.) (v) Without finding Vg, on your...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+4 (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z0 0, 20 2, 204 (Note: Use set notation, and draw a single contour...
Odd and Even Functions An even function has the property f(x) =f(-x). Consider the function f(x)...
Odd and Even Functions An even function has the property f(x) =f(-x). Consider the function f(x) Now, f (-a)-(-a)"-d f(a) An odd function has the property f(-x)-f(x). Consider the function f(x) Now, f (-a) = (-a)' =-a3 =-f(a) Declarative & Procedural Knowledge Comment on the meaning of the definitions of even and odd functions in term of transformations. (i) (ii) Show that functions of the formx) are even. bx2 +c Show, that f(x) = asin xis odd and g(x) =...
1 0 < x < 1, 2. Consider the Haar scaling function, p(x):= { +, and...
1 0 < x < 1, 2. Consider the Haar scaling function, p(x):= { +, and (x) := { -1 10 otherwise 0 0 < x < 1/2, 1/2 < x < 1,. Sketch (by hand is okay) 7(2x) and 7(2x – 1). Show these functions form an orthogonal set. Find the otherwise corresponding orthonormal set. in 3. Let V be a vector space with a complex inner product (,) > Suppose that the set S= {U1, U2, ..., Un}...
Consider the functions f(x)= x? and g(x) = 15x? 20- cb NI (a) Sketch the graph...
Consider the functions f(x)= x? and g(x) = 15x? 20- cb NI (a) Sketch the graph of the pair of functions using a standard window. (b) Describe the transformation used to obtain the graph of the second function from the first function. (a) Use the graphing tool to graph the functions. 8 4- 20 -16 - 12 Loc -4 NH Click to enlarge graph 12 oc 12 16 20 4 -8- -12- - 16- -20-
3. (a) Write a MatLab program that calculates for the function F(x, y) = ln(x + Va,2-y2) The prog...
3. (a) Write a MatLab program that calculates for the function F(x, y) = ln(x + Va,2-y2) The program should use pretty) to display both the original function and the differentiated result, and also use fprintf() to print a label such as "F(x,y) -" and "dF/dxdy - " in front of both the function and the derivative. Then have your program also print out the derivative again after it uses simplify() on the result (b) Find the Taylor expansion of...
Linear Algebra 2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G...
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
2) The set S of all real-valued functions f(x) of a single real variable z is a vector space. (a) Show that the set L of all real-valued linear functions f(x) = mx + b of a single variable x is a subspace of S. (b) Show tha (f(x), g(x))= | f(z)g(x)dx is an inner product on L. (c) Find an orthonormal basis for C with respect to the inner product defined in (b)
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
Question 1. Consider these real-valued functions of two variables: T In (x2 + y2) (a) () What is the maximal domain, D, for the functions f and g? Write D in set notation. (ii) What is the range of f and g? Is either function onto? ii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z00, 2, 04 (Note: Use set notation, and draw a single contour diagram.) (v) Without finding...
Question 1. Consider these real-valued functions of two variables f(x, y) (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? (iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: 20-0, 20-2, 20-4 (Note: Use set notation, and draw a single contour diagram.) (v) Without finding Vg, on your...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+4 (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z0 0, 20 2, 204 (Note: Use set notation, and draw a single contour...
Odd and Even Functions An even function has the property f(x) =f(-x). Consider the function f(x) Now, f (-a)-(-a)"-d f(a) An odd function has the property f(-x)-f(x). Consider the function f(x) Now, f (-a) = (-a)' =-a3 =-f(a) Declarative & Procedural Knowledge Comment on the meaning of the definitions of even and odd functions in term of transformations. (i) (ii) Show that functions of the formx) are even. bx2 +c Show, that f(x) = asin xis odd and g(x) =...
1 0 < x < 1, 2. Consider the Haar scaling function, p(x):= { +, and (x) := { -1 10 otherwise 0 0 < x < 1/2, 1/2 < x < 1,. Sketch (by hand is okay) 7(2x) and 7(2x – 1). Show these functions form an orthogonal set. Find the otherwise corresponding orthonormal set. in 3. Let V be a vector space with a complex inner product (,) > Suppose that the set S= {U1, U2, ..., Un}...
Consider the functions f(x)= x? and g(x) = 15x? 20- cb NI (a) Sketch the graph of the pair of functions using a standard window. (b) Describe the transformation used to obtain the graph of the second function from the first function. (a) Use the graphing tool to graph the functions. 8 4- 20 -16 - 12 Loc -4 NH Click to enlarge graph 12 oc 12 16 20 4 -8- -12- - 16- -20-
3. (a) Write a MatLab program that calculates for the function F(x, y) = ln(x + Va,2-y2) The program should use pretty) to display both the original function and the differentiated result, and also use fprintf() to print a label such as "F(x,y) -" and "dF/dxdy - " in front of both the function and the derivative. Then have your program also print out the derivative again after it uses simplify() on the result (b) Find the Taylor expansion of...
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
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