Applied linear algebra by de Boor C. PDF

By de Boor C.

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Example text

B]) → IR : g → a g(t) dt is linear. , c is the vector space of all convergent sequences. Then the map c → IF : a → limn→∞ an is linear. These examples show that the basic operations in Calculus are linear. This is the reason why so many people outside Algebra, such as Analysts and Applied Mathematicians, are so interested in Linear Algebra. The simplest linear map on a vector space X to a vector space Y is the so-called trivial map. It is the linear map that maps every element of X to 0; it is, itself, denoted by 0.

Thus, IFm×n = L(IFn , IFm ). Thus, a matrix V ∈ IFm×n is associated with two rather different maps: (i) since it is an assignment with domain m × n and values in IF, we could think of it as a map on m × n to IF; (ii) since it is the n-list of its columns, we can think of it as the linear map from IFn to IFm that carries the n-vector a to the m-vector V a = v1 a1 + v2 a2 + · · · + vn an . From now on, we will stick to the second interpretation when we talk about the domain, the range, or the target, of a matrix.

15) A−1 {y} = x + {z ∈ X : Az = 0}. In particular, A is 1-1 if and only if {z ∈ X : Az = 0} = {0}. , ‘at’ 0. For this reason, the set A−1 {0} = {z ∈ X : Az = 0} of all elements of X mapped by A to 0 is singled out. Definition: The set null A := {z ∈ dom A : Az = 0} is called the nullspace or kernel of the linear map A. , contains only the zero vector. The nullspace of a linear map is a linear subspace. Almost all linear subspaces you’ll meet will be of the form ran A or null A for some linear map A.

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Applied linear algebra by de Boor C.

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