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12 Cards in this Set
- Front
- Back
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Euclidean n-space |
the set Rⁿ |
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Addition and scalar multiplication for Euclidean n-space |
(x₁, x₂ ,..., xn) + (y₁, y₂,..., yn) = (x₁ + y₁ , x₂+y₂,..., xn + yn) α(x₁,x₂,...,xn)= (αx₁,αx₂,...αxn) |
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Standard basis vectors of Rⁿ |
e₁=(1,0,0,...,0) e₂=(0,1,0,...,0) e₃=(0,0,1,...,0) continue pattern en=(0,0,0,...,1) |
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Alternate way of writing the vector x=(x₁, x₂ ,..., xn) |
x= x₁e₁ + x₂e₂ + ... + xnen |
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Dot or inner product of x and y in Rⁿ |
x·y= x₁y₁ + x₂y₂ + ... + xnyn |
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Alternate notation for x·y |
<x,y> |
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Length of vector x in Rⁿ |
Length of x= ‖x‖= √(x₁²+ x₂² + ... + xn²) |
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Cauchy-Schwarz Inequality in Rⁿ |
Same as learned earlier |x·y|≤ ‖x‖ ‖y‖ |
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Triangle Inequality in Rⁿ |
‖x+y‖≤ ‖x‖ + ‖y‖ |
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Multiplying n×n matrices (AB=C) |
cij= ∑aikbkj aka multiply row to column |
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Rules for multiplying n×n matrices (AB=C) |
The number of columns in the first matrix must equal the number or rows in the second matrix, if this is not the case, then it is not defined |
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Properties of Matrices |
AB≠BA The only time AB=BA is if it is an identity matrix (I) I*C= C*I =C AA⁻¹=I ABC= (AB)C=A(BC) |
