By Peter Pacheco

Writer Peter Pacheco makes use of an educational method of convey scholars the way to advance powerful parallel courses with MPI, Pthreads, and OpenMP. the 1st undergraduate textual content to without delay deal with compiling and working parallel courses at the new multi-core and cluster structure, An creation to Parallel Programming explains tips to layout, debug, and review the functionality of dispensed and shared-memory courses. undemanding workouts train scholars the best way to bring together, run and regulate instance programs.

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**Example text**

Let {|v 1 , |v 2 , . . }|v n , be a basis for an inner product space V . The Gram-Schmidt process constructs an orthogonal basis |w i as follows: |w1 = |v1 |w2 = |v2 − w1 |v2 |w1 w1 |w1 .. |wn = |vn − w1 |vn w2 |vn wn−1 |vn |w1 − |w2 − · · · − |wn−1 w1 |w1 w2 |w2 wn−1 |wn−1 To form an orthonormal set using the Gram-Schmidt procedure, divide each vector by its norm. For example, the normalized vector we can use to construct |w 2 is |v2 − w1 |v2 |w1 |w2 = |v2 − w1 |v2 |w1 Many readers might find this a bit abstract, so let’s illustrate with a concrete example.

A vector space V is a nonempty set with elements u, v called vectors for which the following two operations are defined: 1. Vector addition: An operation that assigns the sum w = u + v , which is also an element of V ; in other words, w is another vector belonging to the same space 15 VECTOR SPACES 2. Scalar multiplication: Defines multiplication of a vector by a number α such that the vector αu ∈ V In addition the following axioms hold for the vector space V : Axiom 1: Associativity of addition.

4) |a = ⎜ .. ⎠ an This type of notation can be used with qubits. 1). 1. 4), which we call the components of the vector, are complex—something we’ve already mentioned. 5) α|a = α ⎜ .. ⎟ = ⎜ .. ⎠ ⎝ . ⎠ an αan It’s easy to see that this produces another column vector with n complex numbers, so the result is another element in Cn . So Cn is closed under scalar multiplication. 6) |a + |b = ⎜ .. ⎟ + ⎜ .. ⎟ = ⎜ .. ⎠ ⎝ . ⎠ an bn an + bn This should all be pretty straightforward, but let’s illustrate it with an example.