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Vector Operations and Scalar Multiplication of Vactor Assignment
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Assignment Task
1. Given the scalar a and the 5-vectors x abd y state whether the following are valid or invalid.
a. x-2
b. x/y
c. (x,y) (x,x)
d. ax + (xty) 1
e. (1/a)T e1, Where a = 0
2. RGB (red, gren, blue) is a system for represent Colour using a 3 vector where each value ranged form 0 to 255. Use google colour picker to help your answer the following question. You can chage the colour by editing the RGB directly.
We have designed this exercise to be colourblind friendlly but please reach out to us if you feel you are being disadvanteged.
(a) Write out the vector used to represent white. Naw
(b) Describe the colour represented by the vector 255e.
(c) Describe the colour represented by the vector 100e; + 100e3.
(d) Describe the colour represented by the vector 0.
(e) Consider the vector x = (15,100, 155). Describe how the colour of the vector z+al differs from the colour z, where 1
(f) Consider the vector + = (200, 100, 150): Describe the colour represented by the vector. 3. The inner product. is associative with scalar multiplication. This means that (ya)Tb = Y (aTb). (a) Suppose a and 6 are n-vectors. How many flops does it take to calculate (ya)Tb. Show all working. (b) Suppose a and 6 are n-vectors. How many flops does it take to calculate y(aTb). Show all working. (c) Based on your results from part (a) and part (b), w piece of Python code would you use to calculate ya'b? Justify your answer. 4. As we saw in lectures, a vector can be used to represent an image. (a) Colour in a 10 x 10 grid to draw a pixel art alien (each square should be one solid colour). Leave the background of your image white. A gricl has becn provided on page 2 of thé assignment for you to colour in. (b) Suppose that you represent your image as a 100-v) els are denoted with a 0 and non-white colours are represe1. between 1 and 100. What is nnz(X) for your image? 5. You have an affine function f: R3 -> R since this function is affine, it can be written in the form F(x) = aTx + b. You are told that F1 = 0 F(0) = 2, f(e) = 3 and f(z) = -1 for Z = (0,2,1). Using this information determine a and b. 6. Use a counter example to show that the function f(x) = X12 = x2 f : R2 -> R is not linear. 7. Consider the function f(a1,a2) , where a) is given in radians. Find the first order Taylor approximation at the point z = (7/3,1). Calculate f(z) and f(x) at-x = (7/4, 1.25). Give your answers to 2 decimal places. 8. Show that for an n-vector z, and scalars a and B th t 61) a avg(r)+ 3. Formulate your answer using inner products, scalar-vector muftiplication etc. Do not use sigma notaton. This is to help you get some practice with the course notation. 9. Show ny mathematical induction that. 10. On Canvas navigate to 'Modules' in the left menu and select 'Course policies' under 'Unit information'. You must read through and select J understand to every policy on the course o data from Canvas. 11. Led Completion of all exercises in pencast 1. 12. Completion of all exercises in pencast 2. This Mathematics has been solved by our PhD Experts at My Uni Paper.