The operator this particular transformation is a scalar multiplication. The operator is sometimes referred to as what the linear transformation exactly entails.

No, the expected value is not a linear operator, because it isn't an operator (a map from a vector space to itself) at all. The expected value is a linear form, i.e. a linear map from a vector space to its field of …

Mar 28, 2016 · That is, applying the linear operator to each basis vector in turn, then writing the result as a linear combination of the basis vectors gives us the columns of the matrices as those coefficients. …

Feb 27, 2016 · That a linear operator is bounded means simply that it is Lipschitz. It turns out to be equivalent to continuity in case of normed spaces.

Feb 18, 2019 · The zero operator is a linear operator, i.e. a linear map from a vector space to a vector space (possibly the same one). It has the property that it maps any member of the first vector space …

A stronger notion is unitary equivalence, i.e., similarity induced by a unitary transformation (since these are the isometric isomorphisms of Hilbert space), which again cannot happen between a nonunitary …

Sep 17, 2020 · I heard my math teacher mentioning that a matrix is an operator. This confused me, so I looked up what an operator was. In simple terms, it is a function that maps from one space to …

May 19, 2017 · A matrix representation for a linear map describes how the transformation acts in the coordinate space (what you think as an implicit isomorphism is simply the definition).

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This property is unrelated to the completeness of the domain or range, but instead only to the linear nature of the operator. Yes, a linear operator (between normed spaces) is bounded if and only if it is …