1321 Mathematics 2 Flashcards
(28 cards)
What is a Vector?
An element of vector space
What is a vector space?
Defined by the operations you can do to it and properties of these operations
T/F Vector addition is commutative, associative and distributive
True
What is distributivity?
a,b are scalars in R and x,y are vectors
a(x,y) = ax+ay
(a+b)x = ax + bx
What is a linear combination?
Combining vector addition and scalar multiplication
What is the vector space of all polynomials order up to 3 defined over?
The vector [0,1] over the real numbers R
How do we prove a vector space?
Find a zero element
Find an inverse element
And it’s complete if the result of an operation is always a member of the vector space.
What is the generalised theorem of Pythagoras?
sqrt(x^2 + y^2 + z^2) = ||x||
What is a norm?
The length or distance, calculated using the generalised theorem of pythagoras
What does it mean to normalise a vector?
To make it unit length
How can I calculate the scalar product?
<x,y> = x_1 y_1 + x_2 y_2 + x_n y_n
Express norms via a scalar product.
||x|| = <x,x>^1/2
How do we measure angles via the scalar product?
< x,y> = ||x||||y||cos(x,y)
What is the defining property of a unit vector?
||u|| = 1
When are two vectors orthogonal?
When the angle between them is 90* so <x, y> = 0
What is the Cauchty-Schwarx inequality?
|<x,y>| =< ||x||||y||
What is the triangle inequality?
|<x,y>| =< ||x||+||y||
When is the map T linear?
T(ax+by) = aT(x) + bT(y) for all x,y of V and all a,b in R
How do I prove a map isn’t linear?
Provide a counterexample
What is a m by n matrix?
A regular coefficient scheme with m rows and n columns
T/F A matrix is composed of a series of rows and column vectors
True
What is the transpose of a matrix?
Swap the rows and columns
So a 2x3 becomes a 3x2.
What does a matrix represent?
A linear map!
What can matrix vector operations be written as?
Using row vectors and scalar products
