Vectors

Always i, j , k

|a| = amplitude of vector a

i_x j_x i_y j_y

Representing Vectors

R^n

Set of lists of n numbers R^1 is a vector space which is a list of all single numbers

Geometrically

Arrows but without coordinates

Arrows with coordinates

Unit Vectors

(x, y, z) i^ = (1, 0, 0) j^ = (0, 1, 0) k^ = (0, 0, 1)

Cross products

i^ x j^ = +1

$|a\times b|=|a||b|sin(\theta )$

Determinate

Measures how area or volume changes with a linear transformation

[a b] [c d] = ad-bc

Dot products

• Multiply corresponding elements of vector
• Sum resulting coefficients/amplitudes
• Amplitudes that don’t exist are 0, and multiplication rule holds true $a=3j+2k,b=2j$ $a\cdot b=(3\ast 2)$

Other rules

Vectors are orthogonal when their dot product is zero

$a\cdot b=|a||b|cos(\theta )$

$a\cdot a=|a{|}^2$

Matrix Multiplication

• undefined when number of columns of first matrix don’t match number of rows of second matrix

print("[a b] [e f]\n[c d] [g h]")  
a = int(input("a: "))  
b = int(input("b: "))  
c = int(input("c: "))  
d = int(input("d: "))  
e = int(input("e: "))  
f = int(input("f: "))  
g = int(input("g: "))  
h = int(input("h: "))  
alpha = (a*e) + (b*g)  
beta = (a*f) + (b*h)  
carrot = (c*e) + (d*g)  
delta = (c*f) + (d*h)  
print("[", alpha, " ", beta, "]\n[", carrot, " ", delta, "]")

Composition of Transformations

Matrices aren’t associative so you have to do it the right way around $B(A(x))where{M}_1=Aand{M}_{2}isB$

Transposition (X^T)

Generalized Transformations

Counterclockwise R^3 about the x axis

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General or Non-Conforming Rules