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Mathematical Expression Editor
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Essential Vocabulary
Coordinate vector
Let be a vector space, and let be a basis for . If , then the vector in whose
components are the coefficients is said to be the coordinate vector for with respect to
. We denote the coordinate vector by and write:
Finite-dimensional vector space
A vector space is said to be finite-dimensional if it is spanned by finitely many
vectors.
Isomorphism
Let and be vector spaces. If there exists an invertible linear transformation we say
that and are isomorphic and write . The invertible linear transformation is called
an isomorphism.
Matrix of a linear transformation
Let and be finite-dimensional vector spaces with ordered bases and , respectively.
Suppose is a linear transformation.
Then for all vectors in .
Matrix is called the matrix of with respect to ordered bases and .
One-to-one linear transformation
A linear transformation is one-to-one if
Onto linear transformation
A linear transformation is onto if for every element of , there exists an element of
such that .
Subspace
A nonempty subset of a vector space is called a subspace of , provided that is itself
a vector space when given the same addition and scalar multiplication as .
Vector Space
Let be a nonempty set. Suppose that elements of can be added together and
multiplied by scalars. The set , together with operations of addition and scalar
multiplication, is called a vector space provided that
is closed under addition
is closed under scalar multiplication
and the following properties hold for , and in and scalars and :
(a)
Commutative Property of Addition:
(b)
Associative Property of Addition:
(c)
Existence of Additive Identity:
(d)
Existence of Additive Inverse:
(e)
Distributive Property over Vector Addition:
(f)
Distributive Property over Scalar Addition:
(g)
Associative Property for Scalar Multiplication:
(h)
Multiplication by :
We will refer to elements of as vectors.
Note that definitions of span, linear independence, basis, and dimension are
analogous to those for subspaces of .
2024-09-06 02:10:02
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)