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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-
13. The following quadratic function is written in general form: f(x)=x2-6x-7 a) Write the factored form...
13. The following quadratic function is written in general form: f(x)=x2-6x-7 a) Write the factored form b) Write the vertex form (also known as standard form). c) Find the y-intercept. Which of the three forms listed above is the most useful in identifying the y-intercept and why? d) Find the x-intercepts. Which of the three forms listed above is the most useful in identifying the x-intercepts and why? e) Find the vertex. Which of the three forms listed above is...
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-
13. The following quadratic function is written in general form: f(x)=x2-6x-7 a) Write the factored form b) Write the vertex form (also known as standard form). c) Find the y-intercept. Which of the three forms listed above is the most useful in identifying the y-intercept and why? d) Find the x-intercepts. Which of the three forms listed above is the most useful in identifying the x-intercepts and why? e) Find the vertex. Which of the three forms listed above is...
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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