An Introduction to Wavelets Through Linear Algebra by Michael W. Frazier

By Michael W. Frazier

Wavelet conception is at the boundary among arithmetic and engineering, making it excellent for demonstrating to scholars that arithmetic learn is flourishing within the modern-day. scholars can see non-trivial arithmetic rules resulting in average and demanding functions, equivalent to video compression and the numerical resolution of differential equations. the single necessities assumed are a uncomplicated linear algebra history and a bit research history. meant to be as effortless an creation to wavelet conception as attainable, the textual content doesn't declare to be a radical or authoritative reference on wavelet idea.

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4. 26) is the system 1an + Oa21 + 1a31 + 1a21 + Oa31 2an + 1a21 + 3a31 Oan = 3, = 3, = -1, which can be solved, for example, by Gaussian elimination (we omit the calculation), to give an = 13, a21 = 3, and a31 = -10. 27) can be solved to give a 12 = 2, azz = 2, and a32 = -4. Hence A= [ 13 3 -10 2 ] 2 -4 . 49 show that there is a complete correspondence between matrices and linear transformations between finite dimensional vector spaces. Th understand this correspondence better, we consider certain properties that a linear transformation can have and the corresponding properties of the associated matrix.

Wn. First suppose VI, v2 , ... , Vn are linearly independent. Let u E V be arbitrary. •. , Wn}. 12, u, v1. v2 , ... , Vn are linearly dependent. 9(ii), this implies u E span {vi. v2 , ... , Vn}. • , Vn} = V. Now suppose that span {VI, v2 , ... , Vn} = V. If VI, v2 , ... 8(ii), we can find a subset of n - 1 vectors that still spans V. •. 12. This contradiction shows that VI, v2 , ... , Vn are linearly independent. 43 Suppose V is a vector space over a field F and S = {vi, v2 , •.. , Vn} is a basis for V.

The next definition is stated in a general form because it makes sense for any function. 51 Let U and V be sets, and T : U --* V a function. T is one-to-one (written 1 - 1) or injective if T(u 1) = T(u 2) implies u 1 = u 2. Tis onto or surjective if, for every v E V, there exists u E U such that T( u) = v. T is invertible or bijective if T is both 1 - 1 and onto. In other words, Tis 1 - 1 if u1 '# u2 implies T(u 1) '# T(u 2 ); that is, T cannot take two different values to the same value, and T is onto if T attains every element in V.

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