This is a draft cheat sheet. It is a work in progress and is not finished yet.
Complex Numbers
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Polar Form z= rcisθ z= rcisθ
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Modulus ∣z∣= √ a2 + b2 z=a+bi
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Inverses a+b=b+a=0 Additive inverse of -5 is 5 a⋅b=b⋅a=1 Multiplicative inverse of -1 is -1
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Complex Congugate z¯ Flip the sign of the imaginary number to get the conjugate (original a+bi) (complex con a-bi)
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Eulers Identity eiπ+1=0 Eulers equation (eix=cosx+isinx)
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Basis and Dimension
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Linear independence No linear combination of the remaining vectors
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Basis a set of vectors that span a vector space and are linearly independent
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Factor out variables a(1 2 3 1) Number of independent vectors that form a basis
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Dimension of V=dim(V) dimension of Rn is n. Dim(R3)=3
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Rank number of pivots after rref
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Nullity non pivot rows after rref
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tuple 1 column list of numbers
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Dimension of nullspace rref and solve
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Row Reduction
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Augmented Matrix Represents the whole system (line at end)
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RREF Leading 1 then zero under and next leading one beside,only zeros at bottom)
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Augmented RREF rref with complete system
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Gauss-Jordan (elementary row operations) R2=R3....R1=R1-R3....R2=R1-A(R3)
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Determinant
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det(A)= ad-bc row reduce
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Cofactor Expansion Remove row A(3l1) row 3 column 1 you are left with 2x2... then it factors (A31)(2x2matrix)
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det(A)=A31( ad-bc) + A32( ad-bc) + A33( ad-bc) |
Vectorspace
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Subspace set of vectors in W is a subset of the set of vectors in V
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Spanning sets All the matrices that form the same matrix set after
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Linear Transformations
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Kernel rref and solve (1a+0b+3/10c=0 a=-3/)
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Ker(T)= N(A) null space of A |
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Surjection (onto) all outputs could be from 1 input
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Injection (one-to-one) different inputs different outputs
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Bijection (both) both injective and surjective
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Change of Basis t=a,b,c,d v= e,f,g,h....v to t (e,g)=e(ac) + g(bd).... (f,h)=f(a,c) + h(b,d)
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Matrix Multiplication
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Identity Matrix 10 01.....100 010 001
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Elementary matrix is matrix after a elementary row operation |
Inverse & Matrix AlgebrA
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MA=In Left inverse
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AN=In Right inverse
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Inverse of a product inverse all the matrices in set
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Invertible matrices RREF to invert the matrix
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Transpose At= A11 A12 A21 A22 Switch places A12->A21
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Eigenshitazz
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Eignenvalue λIn-A (solve for λ) λ-(a)-b, -c, λ-(d) then det(λ-a) = (λ-a)(λ-d) - (-b)(-c)
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Eigenvector sub in λ to matrix λ-(a)-b, -c, λ-(d) and rref and solve for x's
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Multiplicities eigenspace λ
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Diagonalization can be diagonalized if multiplicities are equal. Needs more than 1 linearly independent eigenvalues
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Need to know
i |
√ -1 |
i2 |
-1 |
multiply 3x1 x 1x3 |
a11 x b11 |
De MOIVRE |
zn=rncis(nθ) |
cis |
cos θ + isin θ or cos θ + (√ -1)sin θ |
det(A) |
ad-bc |
Rn |
range(T) + nullity(T) = n (in m x n) |
m |
row |
n |
column |
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