Unformatted text preview: 8 CHAPTER 2. MATHEMATICAL PREREQUISITES The equivalent of a unit vector for functions is not the function 1, it is the Dirac delta function.
This function will be discussed in detail later, but simply put, it is a single, inﬁnitely high
spike. 2.3 2.3.1 The Dot, oops, INNER Product Solution dota Question: Find the following inner product of the two vectors:
1+i
2−i 2i
3 Answer: Take complex conjugates at the left and sum products of corresponding components
as in
(1 − i) × (2i) + (2 + i) × 3 = 2i + 2 + 6 + 3i = 8 + 5i 2.3.2 Solution dotb Question: Find the length of the vector
1+i
3 Answer: Take the square root of the inner product of the vector with itself:
1+i
3 1+i
3 which works out like
(1 − i)(1 + i) + (3)(3) = √ 1 − i2 + 9 = √ 11 ...
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 Fall '11
 Dr.DanielArenas
 mechanics, Fourier Series, Dot Product, inner product, Dirac delta function

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