Exam (elaborations)
MATH 218 Questions And Answers Latest Update
- Course
- Math 218
- Institution
- Duke University
MATH 218 Questions And Answers Latest Update
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Solution 2024/2025Pepper
MATH 218 Questions And Answers Latest
Update
Transpose
Swap rows and columns
Trace
Sum of entries on the diagonal
Square
Number of rows = number of columns
Lower Triangular
Every entry above diagonal is 0
Upper Triangular
Every entry below diagonal is 0
Diagonal
Every entry not on the diagonal is 0
Zero Matrix
Every entry is 0
Identity
Square matrix with diagonal entries of 1, every other entry is 0
First pivot column
First nonzero column
Rank 1
Every column is a multiple of the first pivot column
, Solution 2024/2025
Pepper
Eigenvector Formula
Av = λV
Euler characteristic
Number of nodes - number of arrows
Connected components
networks of components that form islands
Path
any sequence of distinct arrows joining consecutive nodes (regardless of
direction of arrows)
Cycle
A path that begins and ends at the same node
Circuit Rank
minimum number of arrows that must be removed to break all cycles
Cycle vector
1 traveled with an arrow, -1 traveled against an arrow, 0 did not travel
Netflow
weights in - weights out
Incidence matrix
-1 source node, 1 target node, 0 other nodes
Weight vector
encodes weights of a weighted digraph
Coordinates a vector (geometric)
termination - emination
Length of a vector (geometric)
, Solution 2024/2025
Pepper
√V₁² + V₂²
Arrow scaling thm
length(c * V) = c * length(V)
Unit Vector
a vector with length 1
Vector Normalization
1/length(v) * v
Inner Product
<v,w> = v₁w₁ + v₂w₂ + vₙwₙ
Inner Product (commutative)
<v,w> = <w,v>
Inner Product (left linear)
<c₁v₁ + c₂v₂, w> = c₁<v₁,w> + c₂<v₂,w>
Inner Product (adjoint)
<Av,w> = <v,ATw>
Inner Product (length)
<v,v> = ||v||²
Vector Angles
<v,w> > 0 acute, <v,w> < 0 obtuse, <v,w> = 0, orthogonal
Matrix Multiplication Column Perspective
Matrix Multiplication Row Perspective
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