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Vectors and Matrices Flashcards

1
Q

Define scalar and give some examples

A

A scalar has only magnitude

e.g. Time, speed, voltage, temperature, charge etc

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2
Q

Define a vector and give some examples

A

A vector has both magnitude and direction

e.g. Force, displacement, velocity, acceleration, angular velocity, angular moment, electric field, temperature gradient, magnetic field

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3
Q

Define a matrix

A

A collection of vectors

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4
Q

AB

a = ?

-a = ?

A

a = AB

-a = BA

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5
Q

The magnitude of a vector is the…?

A

…length of the vector

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6
Q

You multiply the vector by a ____ to change the magnitude

A

You multiply the vector by a scalar to change the magnitude

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7
Q

Two vectors are equivalent when…?

A

…both magnitude and direction are the same

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8
Q

Define commutative

A

Same thing but different way round

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9
Q

A(3,1,2) and B(2,3,4)

What would you do to calculate vector AB?

A

You would do B - A so

2-3 = -1

3-1 = 2

4-2 = 2

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10
Q

A(3,1,2) and B(2,3,4)

What would you do to calculate the unit vector along AB?

A

Magnitude of vector AB = √(-12+22+22) = 1/3(-i+j+2k)

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11
Q

It’s in the formula book, but what is the equation for scalar product?

A

a.b = |a|.|b|.cos(ø)

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12
Q

In the scalar product, when ø=0 a.b=?

And when ø = 90˚ a.b=?

A

ø=0 a.b = |a|.|b|

ø = 90˚ a.b = 0

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13
Q

A force moves a mass of 3m in the directino of 2i+5j+3K, find the displacement D

A

3 x unit vector =

3 x x2i+5j+3k / (|2i+5j+3k|) =

3/√38 x (2i+5j+3k)

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14
Q

What is the right hand rule?

A

A rule which determines the orientation of the cross product

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15
Q

What is the equation for angular momentum?

A

d x mv = angular momentum

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16
Q

Two equations for working out the area of a triangle?

A

Half base x height

1/2(|a|.|b|.sinø)

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17
Q

If we could define a matrix multiplcation…(eqn)

A

A.x = B

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18
Q

If A.x = B then

x = ?

A

A.x = B

x = A-1.B

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19
Q

How do you add matrices?

A

You literally just add them!

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20
Q

What type of matrix is this?

00

00

00

A

A 3x2 zero matrix

00

00

00

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21
Q

In matrix power, A4 = ?

A

A4 = A(A(A.A))

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22
Q

(AT)T = ?

(A+B)T = ?

(C.A)T = ?

(A.B)T =

A

(AT)T = A

(A+B)T = AT+BT

(C.A)T = C.AT

(A.B)T = BT.AT

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23
Q

Define a row vector

A

A matrix of dimension 1xn

e.g. A = (a11,a12,a,13…a1n)

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24
Q

Define a column vector

A

A matrix of dimension mx1

e.g. A =

a11,

a21,

a31,

…am1

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25
Define a **square matrix**
Any matrix of dimension **mxn**
26
Define a **symmetric matrix**
Requires aij = aji for all i ≠ j
27
Define a **skew-symmetric matrix**
**Bottom left** is **reflected** from **top** **right** with **opposite** **sign** aij = -aji for all i≠j 2 -1 5 1 0 0 -5 0 3
28
Define **triangular matrix**
**Off digaonal terms** are **zero** in **upper** **right** or **lower** **left** **quadrants**
29
What would a **upper** and **lower** **triangular** **matrix** look like?
**Upper triangular** 1 5 6 0 2 7 0 0 3 **Lower triangular** 1 0 0 4 2 0 5 7 3
30
Define a unit matrix
**Everything** is **0** except the **diagonal terms** 1 0 0 0 1 0 0 0 1
31
When aij = 0 i is what? When aij = 0 i is what?
aij = 0 i = j aij = 0 i ≠ j
32
Unit matrix is sometimes called the...?
...identity matrix
33
What is a **zero matrix**?
With order 3 0 0 0 0 0 0 0 0 0
34
O.A = ? = ?
O.A = O = A.O
35
x1 + x2 = 1 2x1 + 2x2 = 2 Are drawn on a graph, what is the solution?
They are both the same line, so infinite solution
36
x1 + x2 = 1 → ? x1 + x2 = 0 → ?
x1 + x2 = 1 → Inhomogeneous x1 + x2 = 0 → ? Homogeneous
37
x1 + x2 = 1 x1 + x2 = 0 Are drawn on a graph. What is the solution?
No solution because they never touch
38
In Gaussian Elimination and back substiution, A.x = ?
A.x = B
39
Using In Gaussian Elimination and back substiution, If A = 2 5 and B = 2 0 3 -26 What's the answer?
2x1 + 5x2 = 2 3x2 = 26 etc etc
40
Unique solutions, r... Infinite solutions, r... No solutions, r...
Unique solutions, r = n Infinite solutions, r \< n No solutions, r \< m
41
What is a **matrix minor**?
The **determinant** of the **submatrix** formed by deleting the i-th row and j-th column
42
How do you **determine** the **determinant** of a **triangular** **matrix**? -3 0 0 6 4 0 = ? -1 0 2
The determinant is a **product of diagonals** **=** -3 x 4 x 2 = -24
43
What is the determinant if any row (or columns) are 0?
D = 0
44
If you transpose something, what does that do to the determinant?
Nothing
45
If you multiply any row (or column) by scalar R, then the determinant = ?
D = kD
46
Interchanging two rows (or columns) changes signs of only...
...the determinant
47
If two rows (or columns) are identical, then the determinant = ?
D = 0
48
When can you use the **Cramer's Rule**? Also, D(A)...?
Can apply to square matrix A to solve A.x = B for x. Also, D(A) ≠ 0
49
What is the notation for **Cramer's Rule**?
xn = Dn/D
50
In **Cramer's** **Rule**, if the system is **homoegeneous** (\_ = \_) and D≠0 then we get the trivial solution of...?
In **Cramer's** **Rule**, if the system is **homoegeneous** (**_D_** = **_0_**) and D≠0 then we get the trivial solution of... x1 = x2
51
A.x = B → Matrix Inverse → ?
x = A-1.B
52
For an **inverse matrix**, A.A-1 = ? And what is the identity matrix for it?
A.A-1 = I 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
53
When is a matrix full rank?
When the determinant D ≠ 0
54
A-1 = ?
A-1 = 1/ |A|
55
Evaluate the inverse of A = 3 1 2 4
A-1 = 1/10 x 4 -2 -1 3 So you find 1/D and then and swap the diagonals. But the top right and bottom left get multiply by -1
56
**Matrix values** and **eigen** **vectors** **A.x =**?
A.x = lambda.x
57
A.x = lambda.x In this equation, _ will become the eigen values of _ and _ the eigen vectors
A.x = lambda.x In this equation, **_lambda_** will become the eigen values of **_A_** and **_x_** the eigen vectors
58
Would be an eigen vector of 30 ? 40
3 4
59
To find eigen values/vectors we solve...?
A.x = lambda.x A.x - lambda.x = 0 x(A - lambda.I) = 0
60
We want a solution when a determinant = ?
D = 0
61
Eigen vectors are...?
...perpendicular
62
The perpendicular aspect of Eigen systems is...?
...central to solving sets of ODE's
63
We expect linear ODE's...(eqn)
Y = xewt
64
Substitute Y = xewt into... to get...
Substitute Y = xewt into y'' = AY to get w2.x = A.x

Maths (4 decks)

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