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MAT 343 Change of Basis MATLAB A N A PPLICATION OF C HANGE OF B ASIS TO T RIGONOMETRIC F UNCTIONS One of many trigonometric identities is the double angle formula ±²³(´µ) ¶ ´·¸¹ º µ » ¼ . There are similar formulas for all positive integer multiples of x that are obtained by successively applying the addition formula for cosine. We want to let MATLAB find these formulas for us. MATLAB can do symbolic algebra. You just need to tell it that an expression is a symbolic variable: syms x MATLAB functions are vectorized- you can apply them to a whole list of inputs to get a whole list of outputs. 1. Do p = cos([0:5]'*x) to get a vector containing the symbolic expressions ±²³(½µ) for ½ ¶ ¾¿ À À ¿Á . Use expand to perform the symbolic manipulation of expanding each of the functions in p into a sum of powers of cos(x): q=expand(p). Show p and q in your report. Let us now show that the functions in p are linearly independent in the vector space Âþ¿´ÄÅ . To do that, we use the following Theorem: Let Æ Ç ¿ È ¿ Æ É be functions with a common domain Ê Ë Ì . (The set of all such functions forms a vector space V with the usual addition and multiplication). Suppose µ Ç ¿ È ¿ µ É are real numbers in D . Now form the vectors Í Æ Ç Ç ) Î Æ Ç É ) Ï ¿ Í Æ º Ç ) Î Æ º É ) Ï ¿ È ¿ Í Æ É Ç ) Î Æ É É ) Ï If the set of these n vectors is linearly independent in Ì É , then the functions are linearly independent in V. (If the set of n vectors is linearly dependent, then the functions may still linearly dependent or independent, we can’t tell.)
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MAT 343 Change of Basis MATLAB 2. Give an explanation of why this theorem is true. Use the definition of linear independence. Your explanation can be hand-written into your report so you don’t have to type matrices and vectors. 3. If the vectors ( ± ²³ ± ) ´ ± ²³ µ ) ¶ · ( ¸ ²³ ± ) ´ ¸ ²³ µ ) ¶ · ¹ · ( µ ²³ ± ) ´ µ ²³ µ ) are linearly independent, what does that mean about the matrix formed from these vectors? 4. Use what you learned in 1 and 2 to test the functions in p for linear independence. Hint: subs(p,x,2*pi*rand(1)) will plug a random number from the interval º»·¼½¾ into the vector p for x. Use double to convert symbolic into numeric form. Do your findings confirm the independence of the functions in p ? Let S be the subspace of ¿º»·¼½¾ generated by the functions ÀÁ²ó) for Ã Ä »· Å Å ·Æ . We are going to consider two bases of S: Basis V: ºÇ· ÀÁ²³) · ÀÁ²¼³) · ÀÁ²ȳ)· ÀÁ²ɳ) · ÀÁ²Ƴ)¾ Basis U: ºÇ· ÀÁ²³) · ÀÁÂÊ ¸ ²³)· ÀÁÂ Ë ²³)· ÀÁÌ Í ²³)· ÀÁ Π²³)¾ . 5. Find the coordinate vectors of each element of V with respect to the basis U and enter them into MATLAB. Use the result from problem 1. 6. Use the previous problem to find the transition matrix from U to V and use that to express ϲÐ) Ä ÀÁ Π²Ð) Ñ ÀÁÂ Ë ²Ð) as a linear combination of the functions ÀÁ²ó) : a. Write an equation that says that f is a linear combination of the functions ÀÁ²ó) . b. If the relation you found in a. holds for functions, then it also holds for their corresponding U coordinate vectors. Rewrite the equation in terms of U coordinates, and then rewrite one side using matrix-vector multiplication.
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