look at the derivatives evaluated at x 0 ie f i 0 b Show that V is infinite

# Look at the derivatives evaluated at x 0 ie f i 0 b

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, look at the derivatives evaluated at x = 0, i.e. f ( i ) (0).) (b) Show that V is infinite dimensional. (Hint: you need to use the relevant formal definition.) 4. Consider the vector space V of differentiable functions on [0 , 1]. (a) Show that the map A : f ( x ) V f 0 ( x ) + xf ( x ) is a linear operator. (b) Construct the matrix of A with respect to the basis { 1 , x, x 2 , . . . , x n , . . . } . (Since this is an infinite matrix, one needs to give a formula for the i, j el- ement for each pair i, j = 0 , 1 , 2 , . . . .) 5. Show that the collection of Fourier exponential functions v n ( x ) = e inx for i Z forms an orthonormal set in the space of 2 π -periodic functions, given the standard function inner product h f, g i = 1 2 π Z 2 π 0 f * ( x ) g ( x ) dx Provide all details of your calculations. Note that the general complex Fourier series is a vector in the span of this collection. 1

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6. (Change of basis). Suppose we have two bases { ~u i } i =1: N and { ~v i } i =1: N for a vector space, with coordinate operators U , V . (a) Let b be the u -coordinates of any vector ~ b . Show that the v -coordinates b 0 of ~ b are given by b 0 = C u,v b where the change of basis matrix is C u,v = V - 1 U . What is the size of this matrix? (b) Let A and A 0 be the matrix of a linear operator A with respect to the two bases { ~u i } and { ~v i } respectively. Show that A 0 = C u,v A C - 1 u,v . 7. (a) Find the eigenvalues and their eigenspaces, for the rotation matrix R = cos θ sin θ - sin θ cos θ Verify that R is an orthogonal matrix. (b) Find all eigenvalues and a basis for each eigenspace for the matrix 5 / 6 1 / 6 1 / 3 1 / 3 - 1 / 3 1 / 3 1 / 6 5 / 6 - 1 / 3 (Hint: I invite to try out MATLAB on this problem. Use the MATLAB func- tion “eig”. ) 2
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