By Charles E. Leiserson
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Example text
The vector can be written as the linear combination x = x1 e1 + . . + xd ed or, equivalently, as x = x, e1 e1 + . . + x, ed en . 2 Projections The projection of the vector y onto the vector x is the vector yˆ = x, y x. 29) This definition is consistent with a geometrical interpretation of vectors as directed line segments with a common origin. The projection of y onto x is the inner product of the normalized x and y times the normalized x; that is, x ˜, y x ˜, where x ˜ = x/ x . Notice that the order of y and x is the same.
3. Let {vi }ni=1 be an orthonormal basis for the n-dimensional vector space V. Let x ∈ V have the representation bi vi . x= Show that the Fourier coefficients bi can be computed as bi = x, vi . 4. 11); that is, let ρ(x) be defined for the n-vector 2 n |xi | 1/2 ρ(x) = . i=1 Show that ρ(·) is not a norm. 5. 12) and show that the bounds are sharp by exhibiting instances of equality. 6. Prove the following inequalities. a) Prove H¨ older’s inequality: for any p and q such that p ≥ 1 and p + q = pq, and for vectors x and y of the same order, x, y ≤ x p y q.
Ensure that x ˜1 = 0; set x ˜1 = x ˜1 / x ˜1 . 2. If m > 1, for k = 2, . . , m, { for j = k, . . , m, { ˜j − x ˜k−1 , x ˜j x ˜k−1 . 2 Cartesian Geometry } 29 ensure that x ˜k = 0; ˜k / x ˜k . 1 is to be preferred for computations because it is less subject to rounding errors. 1c on page 441. We will not digress here to consider this further, but the difference in the two methods has to do with the relative magnitudes of the quantities in the subtraction. 1 is sometimes called the “modified Gram-Schmidt method”.