FourierMath : Fri Sep 12

• Defined vector addition, talked about components f vectors, representing a vector in terms of different coordinate systems : vector + vector = vector.
• Defined vector dot product: vector "dot" vector = scalar number. In a coordinate sustem, multiply corresponding coordinates and sum.
• Explained (without proof) that perpendicular (also called orthogonal) vectors have dot product zero, gave some examples and tried to build intuition.
• Showed that any vector's dot product with itself is length squared.
• Gave background on what we're trying to accomplish overall, and what this vector stuff has to do with going from time to frequency descriptions of sound.

assignment for next time

• an interactive guide to the Fourier Transform

• vector addition
• vector dot product (also called "inner product")

• What is \( \vec{a} + \vec{b} \) ?
• What is \( \vec{a} \cdot \vec{b} \) ?
• Are these orthogonal? How do you know?
• What is the angle between these vectors?
• Find vectors that are orthogonal to each of \( \vec{a} \) and \( \vec{b} \).

• Find two vectors that are orthogonal to [1,1,1].

• the "T" set : [1,0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1].
• the "F" set : [1,1,1,1], [1,0,-1,0], [0,1,0,-1], [1,-1, 1, -1]

• Show that in each set, all four of these are orthogonal.
• "Normalize" the second set, shrinking them to all have length 1. (The first set already have length 1.)
• Make plots these vectors in IPython, thinking of them as a set of y-values where the x-value are (0,1,2,3).

• Can you write this as a sum of the vectors in set "T", multiplied by some constants?
• Can you write this as a sum of the vectors in set "F", multiplied by some constants?
• Can you always do this for any 4-component vector?

attachments

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Sep12_Ipython_Demo.ipynb Sep 12 2014 11:45 am 18.5kB