Homework SourseConsider the vector space C11 with inner product defined by
Consider the vector space C[-1,1] with inner product defined by (f,g) = f(x)g(x)dx. Find the value of a such that ax + x 3 and 2x ard orthogonal; Show that ||2x||2 + ||3x2||2 = ||2x + 3x2||2.![Consider the vector space C[-1,1] with inner product defined by (f,g) = f(x)g(x)dx. Find the value of a such that ax + x 3 and 2x ard orthogonal; Show that ||2](/WebImages/4/consider-the-vector-space-c11-with-inner-product-defined-by-978854-1761502331-0.webp "Consider the vector space C[-1,1] with inner product defined by (f,g) = f(x)g(x)dx. Find the value of a such that ax + x 3 and 2x ard orthogonal; Show that ||2")
Solution
For ax+x^3 and 2x to be orthogonal we must have
2a =0 or a =0
2) right side by expanding using a+b whole square formula
||2x||^2+||3x^2||^2+2||2x||.||3x^2||
But x and x^2 are orthogonal.
Hence we have right side = left side
