When we measure this qubit in the standard basis, the probability of outcome | 0 \ rangle is | \ alpha | ^ 2 and the probability of outcome | 1 \ rangle is | \ beta | ^ 2.
42.
The permutohedron is a zonotope; a translated copy of the permutohedron can be generated as the Minkowski sum of the " n " ( " n " & minus; 1 ) / 2 line segments that connect the pairs of the standard basis vectors.
43.
If one has a linear transformation T ( x ) in functional form, it is easy to determine the transformation matrix "'A "'by transforming each of the vectors of the standard basis by " T ", then inserting the result into the columns of a matrix.
44.
Thus one can work out both sides of the Cauchy " Binet formula by linearity for every row of " A " and then also every column of " B ", writing each of the rows and columns as a linear combination of standard basis vectors.
45.
We want to give \ mathbb { F } _ q ^ n the standard basis because each coordinate represents a " bit " that is transmitted across a " noisy channel " with some small probability of transmission error ( a binary symmetric channel ).
46.
He made widely cited contributions on the way that fissures and joints in rocks affect the propagation of seismic waves, which has become a standard basis for inferring rock properties in the Earth, and contributed to understanding stressing and deformation in the inflation of the human lung.
47.
If the matrix entries are real numbers, the matrix " A " can be used to represent two linear maps : one that maps the standard basis vectors to the rows of " A ", and one that maps them to the columns of " A ".
48.
The standard basis for R ^ n is the ordered sequence E _ n = \ { e _ 1, \ cdots, e _ n \ }, where e _ j is the element of R ^ n with 1 in the jth place and 0 s elsewhere.
49.
Householder reflections can be used to calculate QR decompositions by reflecting first one column of a matrix onto a multiple of a standard basis vector, calculating the transformation matrix, multiplying it with the original matrix and then recursing down the ( " i ", " i " ) minors of that product.
50.
Investors have noticed in recent weeks that Mexico's one-year cetes don't mature in 364 days, as one-year securities typically do . ( Issuers often use 364 days instead of 365 because that way the " year " is divisible into 13 periods of 28 days, the standard basis for calculating interest payments in Mexico .)
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