Vectors and matrices Flashcards
What is the term when the order of multiplication doesn’t matter
Commutative
What does the dot product end in?
A scalar number
What is the commuatative property?
The order of multiplication doesn’t matter
What is the distributative property?
You can multiply something outside of a bracket to the components inside
Are vectors distributive?
Yes
What is it called when you use dot product and multiple a vector over parenteses?
Distributive
What is associative?
The grouping of multiplication doesn’t matter
What is the difference between commutative, distributive and associative?
Commutative - the order of multiplication doesn’t matter
Distributive - you can multiply something outside of the bracket to the inside components
Associative - the grouping of multiplication doesn’t matter
Whats the notation for a vector?
lower case with an arrow above it
What is R2
Ordered 2 tupes where all the vlaues are real numbers
What is standard position?
Origin
What is the relationship between x-y and y - x
The same magnitude but opposite directions
How do you denote a unit vector?
A hat
What is a hat and arrow for?
Hat is unit vector
Arrow is a normal vector
What are position vectors?
Tehy are co-ordinates i.e. they start at the origin
What is the magnitude of a vector?
It’s the length of the line
How do you measure the magnitude of a vector?
Create triangles and use pythagorus
How do you denomate the magniude of a vector?
|| a with arrow ||
How do you find a unit vector with a given direction?
Divide by the magnitude of the vector (so that the magnitude is 1)
How do you represent a line? Where the line goes through the origin?
V = [2 1]
S = { c v | E R}
Set where scalar of the vector v where the scalar is a real number
How do you get the vector between 2 position vectors?
Subtract one from the other
Does the order matter when working out the vector between 2 position vectors
No
How do you document a line that does not go through the origin?
S = { x + c v | c E R}
How do you document a line that does not go through the origin and is given by 2 position vectors?
S = {a + c ( b - a) | c E R}
True or false: can you reach any point on a 2d with linear combinations?
True
What is linear independence?
Vectors that cannot be expressed by the other ones
What do you have to have to represent any point in R2?
Two linerally indepednet vectors
How do you prove that a set of vectors are lineally indepednet?
If you have any non zero value scalar where the vectors can be combined to be [0 0]
How do you get the modulus of a vector?
Dot product of itself square rooted
What is r.r?
r.r = ∥r∥2
What is r.s?
r.s = ∥r∥∥s∥ cos θ
What does it mean when r.s is zero?
They are orthogonal
What does it mean when r.s is negative?
They are going in opposite directions
What does it mean when r.s is positive?
They are going in the same direction
What is the scalar projection?
Projection
The component of a vector a that is in the same direction as of vector b (Hence projection is a vector) Length of the projection does not depend on the length(magnitude) of b. See the image below
What is the forumula for scalar projection?
scalar projection:
r.s
_
∥r∥
What is the vector projection?
The sclar projecttion times by a unit vector in the direction of r
What is the fomula for vector projection?
r.s
_
r.r
x
r
How do you change the basis in new orthogonal vectors?
For R in b1 and b2:
r. b1 / b1.b1
r. b2 / b2.b2
What is basis?
Basis is a set of n vectors that are linearly independent and span the space
What do basis vector not have to be?
unit vectors or orthogonal
What matrix changes nothing?
The identity matrix
1 0
0 1
What does this matrix do?
0 1
1. 0
Mirror at 45 degrees
What does this matrix do?
0 -1
-1 0
Mirror at 320 degrees
What does this matrix do?
-1 0
0 1
Mirror along the vertical axis
What does this matrix do?
1 0
0 - 1
Mirror along the horizontal axis
What does this matrix do?
0 1
1 0
Rotate by 90 degrees anticlockwise
What is the general equation for a rotation?
cos theta sin theta
-sin theta cos theta
Are matrices commutative?
No
Are matrices associative?
Yes
What is the determinant?
The area in space that has got bigger by a factor
How do you denote the determinant?
A |
What is | A | ?
ad - bc where A
a b
c d
How do you find the inverse of a 2 x 2 matrix?
a b
c d
d -b
c -a
What is the determinant for
1 2
1 2
0 - it’s on a line
