Vectors and Matrices Flashcards
Define scalar and give some examples
A scalar has only magnitude
e.g. Time, speed, voltage, temperature, charge etc
Define a vector and give some examples
A vector has both magnitude and direction
e.g. Force, displacement, velocity, acceleration, angular velocity, angular moment, electric field, temperature gradient, magnetic field
Define a matrix
A collection of vectors
AB
a = ?
-a = ?
a = AB
-a = BA
The magnitude of a vector is the…?
…length of the vector
You multiply the vector by a ____ to change the magnitude
You multiply the vector by a scalar to change the magnitude
Two vectors are equivalent when…?
…both magnitude and direction are the same
Define commutative
Same thing but different way round
A(3,1,2) and B(2,3,4)
What would you do to calculate vector AB?
You would do B - A so
2-3 = -1
3-1 = 2
4-2 = 2
A(3,1,2) and B(2,3,4)
What would you do to calculate the unit vector along AB?
Magnitude of vector AB = √(-12+22+22) = 1/3(-i+j+2k)
It’s in the formula book, but what is the equation for scalar product?
a.b = |a|.|b|.cos(ø)
In the scalar product, when ø=0 a.b=?
And when ø = 90˚ a.b=?
ø=0 a.b = |a|.|b|
ø = 90˚ a.b = 0
A force moves a mass of 3m in the directino of 2i+5j+3K, find the displacement D
3 x unit vector =
3 x x2i+5j+3k / (|2i+5j+3k|) =
3/√38 x (2i+5j+3k)
What is the right hand rule?
A rule which determines the orientation of the cross product
What is the equation for angular momentum?
d x mv = angular momentum
Two equations for working out the area of a triangle?
Half base x height
1/2(|a|.|b|.sinø)
If we could define a matrix multiplcation…(eqn)
A.x = B
If A.x = B then
x = ?
A.x = B
x = A-1.B
How do you add matrices?
You literally just add them!
What type of matrix is this?
00
00
00
A 3x2 zero matrix
00
00
00
In matrix power, A4 = ?
A4 = A(A(A.A))
(AT)T = ?
(A+B)T = ?
(C.A)T = ?
(A.B)T =
(AT)T = A
(A+B)T = AT+BT
(C.A)T = C.AT
(A.B)T = BT.AT
Define a row vector
A matrix of dimension 1xn
e.g. A = (a11,a12,a,13…a1n)
Define a column vector
A matrix of dimension mx1
e.g. A =
a11,
a21,
a31,
…am1
Define a square matrix
Any matrix of dimension mxn
Define a symmetric matrix
Requires aij = aji for all i ≠ j
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
Define triangular matrix
Off digaonal terms are zero in upper right or lower left quadrants
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
Define a unit matrix
Everything is 0 except the diagonal terms
1 0 0
0 1 0
0 0 1
When aij = 0 i is what?
When aij = 0 i is what?
aij = 0 i = j
aij = 0 i ≠ j
Unit matrix is sometimes called the…?
…identity matrix
What is a zero matrix?
With order 3
0 0 0
0 0 0
0 0 0
O.A = ? = ?
O.A = O = A.O
x1 + x2 = 1
2x1 + 2x2 = 2
Are drawn on a graph, what is the solution?
They are both the same line, so infinite solution
x1 + x2 = 1 → ?
x1 + x2 = 0 → ?
x1 + x2 = 1 → Inhomogeneous
x1 + x2 = 0 → ? Homogeneous
x1 + x2 = 1
x1 + x2 = 0
Are drawn on a graph. What is the solution?
No solution because they never touch
In Gaussian Elimination and back substiution,
A.x = ?
A.x = B
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
Unique solutions, r…
Infinite solutions, r…
No solutions, r…
Unique solutions, r = n
Infinite solutions, r < n
No solutions, r < m
What is a matrix minor?
The determinant of the submatrix formed by deleting the i-th row and j-th column
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
What is the determinant if any row (or columns) are 0?
D = 0
If you transpose something, what does that do to the determinant?
Nothing
If you multiply any row (or column) by scalar R, then the determinant = ?
D = kD
Interchanging two rows (or columns) changes signs of only…
…the determinant
If two rows (or columns) are identical, then the determinant = ?
D = 0
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
What is the notation for Cramer’s Rule?
xn = Dn/D
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
A.x = B → Matrix Inverse → ?
x = A-1.B
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
When is a matrix full rank?
When the determinant D ≠ 0
A-1 = ?
A-1 = 1/ |A|
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
Matrix values and eigen vectors
A.x =?
A.x = lambda.x
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
Would be an eigen vector of 30 ?
40
3
4
To find eigen values/vectors we solve…?
A.x = lambda.x
A.x - lambda.x = 0
x(A - lambda.I) = 0
We want a solution when a determinant = ?
D = 0
Eigen vectors are…?
…perpendicular
The perpendicular aspect of Eigen systems is…?
…central to solving sets of ODE’s
We expect linear ODE’s…(eqn)
Y = xewt
Substitute Y = xewt into… to get…
Substitute Y = xewt into y’’ = AY to get
w2.x = A.x
