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2. Use Gram-Schmidt process to find an orthonormal basis for the set of three functions $1, x, x^3$ given the inner product is defined as $<f, g> = int_{-1}^{1} f(x)g(x)dx.$

          2. Use Gram-Schmidt process to find an orthonormal basis for the set of three functions $1, x, x^3$ given the inner product is defined as $<f, g> = int_{-1}^{1} f(x)g(x)dx.$
        
2. Use Gram-Schmidt process to find an orthonormal basis for the set of three functions 1, x, x^3 given the inner product is defined as <f, g> = int-1^1 f(x)g(x)dx.

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Use the Gram-Schmidt process to find an orthonormal basis for the set of three functions 1, x, and x^3, given that the inner product is defined as <f, g> = ∫ f(x)g(x)dx.
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Transcript

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00:01 Hi, now we are going to find the arthonormal basis.
00:04 The given standard bases are 1 comma x.
00:08 X cube and the given condition is inner product f comma g is equal to integral over minus 1 to 1 to 1 f of x into g of x into dx.
00:28 Now i take v1 is equal to 1 and v2 is equal to x and v3 is equal to x cube.
00:40 W1 is equal to v1.
00:43 So we have w1 is equal to 1.
00:48 And next we are going to calculate w2.
00:52 W2 is equal to v2 minus inner product.
00:59 V2 comma w1 divide by norm of w1 into w1.
01:09 Now we are going to calculate inner product v2 comma w1 and it will be equal to inner product x comma 1 that is integral over minus 1 to 1 x d x.
01:30 After integration and substituting the limits, it will be equal to 0.
01:36 And next, norm of w1 is equal to integral over minus 1 to 1 -d -x and it will be equal to 2.
01:52 Now substitute the corresponding values in the w -2 formula.
01:56 Then we have w -2 is equal to x minus 0 divide by 2...
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