Week 15

Question 1

For symmetrical matrix, how to get matrix A B->B

Answer 1

• if the base belongs to the transformation e.g. (the Range) , multiply by the eigenvalue

-for kernel we multiply the eigenvector of kernel by it’s eigenvalue

Question 2

∇ f( x,y) (gradient of f)

Answer 2

-partial derivatives of each, sub in then transposed

Question 3

slope of d?

Answer 3

vertical displacement/ horizontal distance

tranpose the unit vector and divide and we get slope

or directional derivative of terms of d except p.d at point (a,b,c)

Question 4

Unit vector when gradient is same direction as directional derivative (rate of increase is max)

Answer 4

u = gradient of f / mod gradient of f

Question 5

Unit vector when gradient is opposite direction as directional derivative (rate of decrease is max)

Answer 5

u = - gradient of f / mod gradient of f

Question 6

Unit vector when gradient is opposite direction as directional derivative (no rate)

Answer 6

negative the gradient

Question 7

Directional derivative (slope of d (A vector))?

Answer 7

<Û ,∇ f(a,b)>

where Û = unit vector in the directon u= (a , b)

Question 8

Contour map?

Answer 8

when you make f(x1,x2) = c , jus mapped onto one 2d space

bunch of horizontal sections

Question 9

Vertical sections?

Answer 9

making x3= f(x1,a) := g(x1)

Question 10

Partial derivative notation?

Answer 10

fx1 for p.d f(x1,x2)

Question 11

Ordinary derivative (what is it)?

Answer 11

gives the slope of intersection between f(x1,x2) = x3 and x2=b

measure at a in x1 +ve direction

Question 12

Ordinary derivative eq?

Answer 12

fx(a,b) = g’(a)

where g(x1) = f(x1,b)

Question 13

Direction vector at a for the tangent line to the curve when x2=b?

Answer 13

(1 )
(0 )
(fx(a,b))

Question 14

Direction vector at b for the tangent line to the curve when x2=a?

Answer 14

(0 )
(1 )
(fy(a,b))

Question 15

Tangent plane eq?

Answer 15

x3 - f(a,b) = fx1(a,b)(x1-a) + fx2(a,b)(x2-b)

Question 16

Normal to tangent plane?

Answer 16

(fx(a,b) )
(fy(a,b) )
(-1 )

Question 17

Way to denote tangent plane (matrix)?

Answer 17

x3 - f(a) = f’(a)(x-a)

at a point in x3 = (a,f(a))

where (x-a) = (x1-a )
(x2 - b)

and f’(a) = (fx(a,b) fy(a,b) «derivative of f , transpose is gradient of f

Question 18

What does gradient of f tell us ∇ f( x,y)?

Answer 18

Orthogonal to contour at f(a,b) and going towards increasing contour values

Question 19

General vector (d) for tangent plane)?

Answer 19

orthogonal to normal vector

Question 20

Other way to denote directional derivaitve and what does it mean?

Answer 20

ll Û ll ll∇ f(a,b)ll cos θ

where theta is angle between direction of unit vector and gradient of f

max when in the same direction, min when opposite and 0 when orthogonal to gradient

Question 21

Eigenvalue x eigenvector?

Answer 21

gives linear transformation e.g relfection in y=3x with eigenvector (1) with value 1
(3)

then value -1 with vector (-3)
(1) <as orthogonal

Question 22

How to show homogenous?

Answer 22

use lambda then power of lambda gives you degree

Question 23

When tangent plane is horizontal?

Answer 23

partial derivatives are zero (gradient) and z = f(a,b)

so we run sim equations to get a,b f(a,b)

Question 24

Parametric eq for tangent plane?

Answer 24

particular vector is point (a,b,c) and then direction vectors are just (1,0,p.d x (a)) and (0,1,p.d y(b))

Question 25

Implicit differentiation, to find slope of contour (if not given eq for contour)?

Answer 25

change f(a,b) = f(x1,g(x1)) so x2 becomes x2(x1)

Question 26

Top of partial derivate eq derivation thing?

Answer 26

where (a,b) = (3,3) using 3+t for both if u= (1,1) if (1,0) we just use it for one and bottom we just use magnitude of u x t

Question 27

Parametric eq for tangent line?

Answer 27

(a ) (fx2(a,b)) (b ) + λ (-fx1(a,b)) (f(a,b) ( 0 )

Question 28

Direction vector if looking for a vertical plane in R^3?

Answer 28

(0) (0) (1)