Download MATH00040 Formula Sheet Matrices and Vectors Matrix inverses: If

MATH00040 Formula Sheet
Matrices and Vectors
a b
Matrix inverses: If ad − bc 6= 0, then
c d
!−1
=
Determinants:
det

a11 a12 a13
a11 a12
a21 a22

!
a
11 a12
=
a21 a22
a


22 a23


det  a21 a22 a23  = a11 a32 a33
a31 a32 a33
1
ad − bc
d −b
−c
a
!
= a11 a22 − a12 a21 .
a
21 a23
− a12 a31 a33
.
a
21 a22
+ a13 a31 a32
.
p
Length of a vector: If a = (a1 , a2 , a3 ), then kak = a21 + a22 + a23 .
Dot product: If a = (a1 , a2 , a3 ) and
, b3 ), then a · b = a1 b1 + a2 b2 + a3 b3 .
b = (b1 , b2
v
·
w
.
Angle between vectors: θ = cos−1
kvk· kwk
î ĵ k̂ Cross product: v × w = v1 v2 v3 , where v = (v1 , v2 , v3 ) and w = (w1 , w2 , w3 ).
w1 w2 w3 Eigenvalues: To find the eigenvalues of A, solve det(A − λI) = 0.
Eigenvectors: To! find the eigenvector v corresponding to the eigenvalue λ, solve Av = λv,
x
where v =
.
y
Complex numbers
Real part of a complex number: Re(a + bi) = a.
Imaginary part of a complex number: Im(a + bi) = b.
√
Modulus of a complex number: |a + bi| = a2 + b2 .
Complex conjugate of a complex number: a + bi = a − bi.
a + bi
a + bi c − di
Dividing complex numbers:
=
·
.
c + di
c + di c − di
c
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Polar form: a + bi = r(cos(θ)
+i sin(θ)), where r = |a + bi| and we find θ as follows.
b
First calculate φ = tan−1 . Then
a
θ = φ if a > 0 and b > 0,
θ = φ − π if a < 0 and b < 0
and
θ = π − φ if a < 0 and b > 0,
θ = −φ if a > 0 and b < 0.
Common values of tan−1 :
−1
tan (0) = 0,
−1
tan
1
√
3
=
π
,
6
tan−1 (1) =
π
,
4
tan−1
√ π
3 = .
3
De Moivre’s formula: [r(cos(θ) + i sin(θ))]n = r n (cos(nθ) + i sin(nθ)).
Roots of complex numbers: The n’th roots of r(cos(θ) + i sin(θ)) are
1
θ 2kπ
θ 2kπ
+ i sin
k = 0, 1, . . . , n − 1.
zk = r n cos
+
+
n
n
n
n
Differential Calculus
f (x)
f ′ (x)
c
0
xn
nxn−1
eax
aeax
1
x
a cos(ax)
ln(ax)
sin(ax)
cos(ax)
−a sin(ax)
Comments
Here c is any real number
Here we must have ax > 0
Note the change of sign
Table 1: Some common derivatives
′
The product rule for differentiation: (f
= f ′ (x)g(x) + f (x)g ′ (x).
g)(x)
′
f ′ (x)g(x) − f (x)g ′(x)
f
.
(x) =
The quotient rule for differentiation:
g
g 2 (x)
dy du
dy
=
· .
The chain rule for differentiation:
dx
du dx
The critical points of a function f occur at points x where f ′ (x)
√ = 0.
−b ± b2 − 4ac
The solutions of the equation ax2 + bx + c = 0 are x =
.
2a
′′
Classifying critical points: Local minima occur where f (x) > 0.
Local maxima occur where f ′′ (x) < 0.
f (xn )
.
The Newton-Raphson method: xn+1 = xn − ′
f (xn )
c
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Integral Calculus
Z
f (x)
k
f (x) dx
kx + c
1
xn+1 + c
n+1
ln(x) + c
1 ax
e +c
a
1
− cos(ax) + c
a
1
sin(ax) + c
a
xn
1
x
eax
sin(ax)
cos(ax)
Comments
Here k is any real number
Here we must have n 6= −1
Here we must have x > 0
Note the change of sign
Table 2: Some common integrals
Evaluating definite integrals: ZIf F is an antiderivative of f then
b
f (x) dx = [F (x)]ba = F (b) − F (a).
a
du
Integration by substitution: dx =
.
du/dx
Z
Z
′
Integration by parts:
f (x)g (x) dx = f (x)g(x) − f ′ (x)g(x) dx.
Z b
Volume of solid of revolution: V = π
f (x)2 dx.
a
Probability
Sum rule for probability: P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
P (A ∩ B)
.
Conditional probability: P (A|B) =
P(B)
n k
p (1 − p)n−k .
Binomial distribution: P (X = k) =
k
λk e−λ
Poisson distribution: P (X = k) =
.
k! a−µ
b−µ
Normal distribution: P (a 6 X 6 b) = P
.
6Z6
σ
σ
c
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