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Linear Algebra - MT2 Cheat Sheet by

This is for math 232 - linear algebra, midterm 2

Matrix Algebra

AT = A
(kA)T = k(A)T
(A+B)T = AT + BT
(AB)T = BTAT
tr(AT) = tr(A)
tr(AB) = tr(BA)
uTv = tr(uvT)
tr(uvT) = tr(vuT)
(AT)ij = Aji
tr(A) = a11 + a22 ... + ann

Elementary Matrix

elementary matrices are invertible
- A -> RREF = I
- A can be express as a product of E
- A is invertible
- Ax = 0 has only the trivial solution
- Ax = b is consistent for every vector b in Rn
- Ax = b has eactly 1 solution for every b in Rn
If not, then all no.
A-1 = Ek Ek-1 ...  E2 E1
[ A | I ] -> [ I | A-1 ]
(how to find inverse of A)
Ax = b; x = A-1b

Consis­tency

EAx = Eb -> Rx = b' , where b' = Eb
(Ax=b) [ A | b ] -> [ EA | Eb ] (Rx = b')
(but treat b as unknown: b1, b2...)
For it to be consis­tent, if R has zero rows at the bottom, b' that row must equal to zero

Homoge­neous Systems

Linear Combin­ation of the vectors:
v = c1v1 + c2v2 ... + cnvn
(use matrix to find c)
Ax = 0
Homoge­neous
Ax = b
Non-ho­mog­enous
x = x0 + t1v1 + t2v2 ... + tkvk
Homoge­neous
x = t1v1 + t2v2 ... + tkvk
Non-ho­mog­eneous
xp is any solution of NH system
and xh is a solution of H system
x = xp + xh

Determ­inants

det(A) = a1jC1j + a2jC2j ... + anjCnj
expansion along jth column
det(A) = ai1Ci1 + ai2Ci2 ... + ainCin
expansion along the ith row
Cij = (-1)i+j Mij
Mij = deleted ith row and jth column matrix
- pick the one with most zeros to calculate easier
det(AT) = det(A)
det(A-1) = 1/det(A)
det(AB) = det(A)­det(B)
det(kA) = kndet(A)
- A is invertible iff det(A) not equal 0
- det of triangular or diagonal matrix is the product of the diagonal entries
det(A) for 2x2 matrix
ad - bc
 

Examples of Subspaces

IF: w1, w2 are within S
then w1+w2 are within S
and kw1 is within S
- the zero vector 0 it self is a subspace
- Rn is a subspace of all vectors
- Lines and planes through the origin are subspaces
- The set of all vectors b such that Ax = b is consis­tent, is a subspace
- If {v1, v2, ...vk} is any set of vectors in Rn, then the set W of all linear combin­ations of these vector is a subspace
W = {c1v1 + c2v2 + ... ckvk}; c are within real numbers

Span

- the span of a set of vectors { v1, v2, ... vk} is the set of all linear combin­ations of these vectors
span { v1, v2, ... vk} = { v11t, t2v2, ... , tkvk}
If S = { v1, v2, ... vk}, then W = span(S) is a subspace
Ax = b is consistent if and only if b is a linear combin­ation of col(A)

Linear Indepe­ndent

- if unique solution for a set of vectors, then it is linearly indepe­ndent
c1v1 + c2v2 ... + cnvn = 0; all the c = 0
- for dependent, not all the c = 0
Dependent if:
- a linear combin­ation of the other vectors
- a scalar multiple of the other
- a set of more than n vectors in Rn
Indepe­ndent if:
- the span of these two vectors form a plane
- list the vectors as the columns of a matrix, row reduce it, if many solution, then it is dependent
- after RREF, the columns with leading 1's are a maxmially linearly indepe­ndent subset according to Pivot Theorem

Diagonal, Triang­ular, Symmetric Matrices

Diagonal Matrices
all zeros along the diagonal
Lower Triangular
zeros above diagonal
Upper Triangular
zeros below the diagonal
Symmetric if:
AT = A
Skew-S­ymm­etric if:
AT = -A

Adjoint and Cramer's Rule

adj(A) = CT
CT = matrix confactor of A
A-1 = (1/det(A)) adj(A)
adj(A)A = det(A) I
x1 = det(A1) / det(A)
x2 = det(A2) / det(A)
xn = det(An) / det(A)
det(A) not equal 0
An is the matrix when the nth column is replaced by b
 

Hyperp­lane, Area/V­olume

a hyperplane in Rn
a1x1 + a2x2 ... + anxn = b
- can also written as ax = b
to find aperp
ax = 0, find the span
if A is 2x2 matrix:
- |det(A)| is the area of parall­elogram
if A is 3x3 matrix:
- |det(A)| is the volume of parall­ele­piped
- subtract points to get three vectors, then make it to a matrix to find the area/v­olume

Cross Product

u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)
u x v = -v x u
k(u x v) = (ku) x v = u x (kv)
u x u = 0
parallel vectors has 0 for c.p.
u (u x v) = 0
v (u x v) = 0
u x v is perpen­dicular to span {u, v}
||u x v|| = ||u|| ||v|| sin(th­eta), where theta is the angle between vectors

Complex Number

complex number
a + ib
(a + ib) + (c + id) = (a + c) + i(b + d)
(a + ib) - (c + id) = (a - c) + i(b - d)
(a + ib) (c + id) = (ac + bd) + i(ad + bc)
(a + bx) (c + dx) = (ac + bdx2) + x(ad + bc)
i2 = -1
z = a + ib
z bar = a - ib
the length­(ma­gni­tude) of vector z
|z| = sqrt(z x z bar)
= sqrt­(a2 + b2)
z-1 = 1/z = z bar / |z|2
z1 / z2 = z1z2-1
z = |z| (cos(θ) + i (sin(θ))
polar form (r = |z|)
z1z2 = |z1| |z2| (cos(θ1 + θ2) + i (sin(θ1 + θ2))
z1/z2 = |z1| / |z2| (cos(θ1 - θ2) + i (sin(θ1 - θ2))
zn = rn(cos(n θ) + i sin(n θ))
r = |z|
ei theta = cos(θ) + i sin(θ)
ei pi = -1
ei pi +1 = 0
z1z2 = r1r2 ei (θ1 + θ2)
zn = rn ei nθ
z1 /z2 = r1 / r2 ei (θ1 - θ2)

Eigenv­alues and Eigenv­ectors

Ax= λx
det(λI - A) = (-1)n det(A - λI)
pa(λ) = 3x3: det(A - λI); 2x2: det(λI - A)
- solve for (λI - A)x = 0 for eigenv­ectors
Work Flow:
- form matrix
- compute pa(λ) = det(λI - A)
- find roots of pa(λ) -> eigenv­alues of A
- plug in roots then solve for the equation
 

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