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Linear algebra > Vectors and matrices > Flashcards

Vectors and matrices Flashcards

1
Q

What is the term when the order of multiplication doesn’t matter

A

Commutative

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2
Q

What does the dot product end in?

A

A scalar number

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3
Q

What is the commuatative property?

A

The order of multiplication doesn’t matter

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4
Q

What is the distributative property?

A

You can multiply something outside of a bracket to the components inside

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5
Q

Are vectors distributive?

A

Yes

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6
Q

What is it called when you use dot product and multiple a vector over parenteses?

A

Distributive

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7
Q

What is associative?

A

The grouping of multiplication doesn’t matter

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8
Q

What is the difference between commutative, distributive and associative?

A

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

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9
Q

Whats the notation for a vector?

A

lower case with an arrow above it

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10
Q

What is R2

A

Ordered 2 tupes where all the vlaues are real numbers

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11
Q

What is standard position?

A

Origin

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12
Q

What is the relationship between x-y and y - x

A

The same magnitude but opposite directions

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13
Q

How do you denote a unit vector?

A

A hat

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14
Q

What is a hat and arrow for?

A

Hat is unit vector

Arrow is a normal vector

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15
Q

What are position vectors?

A

Tehy are co-ordinates i.e. they start at the origin

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16
Q

What is the magnitude of a vector?

A

It’s the length of the line

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17
Q

How do you measure the magnitude of a vector?

A

Create triangles and use pythagorus

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18
Q

How do you denomate the magniude of a vector?

A

|| a with arrow ||

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19
Q

How do you find a unit vector with a given direction?

A

Divide by the magnitude of the vector (so that the magnitude is 1)

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20
Q

How do you represent a line? Where the line goes through the origin?

A

V = [2 1]

S = { c v | E R}

Set where scalar of the vector v where the scalar is a real number

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21
Q

How do you get the vector between 2 position vectors?

A

Subtract one from the other

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22
Q

Does the order matter when working out the vector between 2 position vectors

A

No

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23
Q

How do you document a line that does not go through the origin?

A

S = { x + c v | c E R}

24
Q

How do you document a line that does not go through the origin and is given by 2 position vectors?

A

S = {a + c ( b - a) | c E R}

25
Q

True or false: can you reach any point on a 2d with linear combinations?

A

True

26
Q

What is linear independence?

A

Vectors that cannot be expressed by the other ones

27
Q

What do you have to have to represent any point in R2?

A

Two linerally indepednet vectors

28
Q

How do you prove that a set of vectors are lineally indepednet?

A

If you have any non zero value scalar where the vectors can be combined to be [0 0]

29
Q

How do you get the modulus of a vector?

A

Dot product of itself square rooted

30
Q

What is r.r?

A

r.r = ∥r∥2

31
Q

What is r.s?

A

r.s = ∥r∥∥s∥ cos θ

32
Q

What does it mean when r.s is zero?

A

They are orthogonal

33
Q

What does it mean when r.s is negative?

A

They are going in opposite directions

34
Q

What does it mean when r.s is positive?

A

They are going in the same direction

35
Q

What is the scalar projection?

A

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

36
Q

What is the forumula for scalar projection?

A

scalar projection:
r.s
_
∥r∥

37
Q

What is the vector projection?

A

The sclar projecttion times by a unit vector in the direction of r

38
Q

What is the fomula for vector projection?

A

r.s

_

r.r

x

r

39
Q

How do you change the basis in new orthogonal vectors?

A

For R in b1 and b2:

r. b1 / b1.b1
r. b2 / b2.b2

40
Q

What is basis?

A

Basis is a set of n vectors that are linearly independent and span the space

41
Q

What do basis vector not have to be?

A

unit vectors or orthogonal

42
Q

What matrix changes nothing?

A

The identity matrix
1 0
0 1

43
Q

What does this matrix do?
0 1
1. 0

A

Mirror at 45 degrees

44
Q

What does this matrix do?
0 -1
-1 0

A

Mirror at 320 degrees

45
Q

What does this matrix do?
-1 0
0 1

A

Mirror along the vertical axis

46
Q

What does this matrix do?
1 0
0 - 1

A

Mirror along the horizontal axis

47
Q

What does this matrix do?
0 1
1 0

A

Rotate by 90 degrees anticlockwise

48
Q

What is the general equation for a rotation?

A

cos theta sin theta

-sin theta cos theta

49
Q

Are matrices commutative?

A

No

50
Q

Are matrices associative?

A

Yes

51
Q

What is the determinant?

A

The area in space that has got bigger by a factor

52
Q

How do you denote the determinant?

A

A |

53
Q

What is | A | ?

A

ad - bc where A

a b
c d

54
Q

How do you find the inverse of a 2 x 2 matrix?

a b
c d

A

d -b

c -a

55
Q

What is the determinant for
1 2
1 2

A

0 - it’s on a line

Linear algebra (1 decks)

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