Linear Algebra

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Vector spaces

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A vector space( or linear space) V over a number field² F consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so, that for each pair of elements x, y, in V there is a unique element x + y in V, and for each element a in F and each element x in V there is a unique element ax in V, such that the following conditions hold.
(VS 1) For all in V, (commutativity of addition).
(VS 2) For all in V, (associativity of addition).
(VS 3) There exists an element in V denoted by such that for each in V.
(VS 4) For each element in V there exists an element in V such that .
(VS 5) For each element in V, .
(VS 6) For each pair of element a in F and each pair of elements in V, .
(VS 7) For each element in F and each pair of elements in V, .
(VS 8) For each pair of elements in F and each pair of elements in V, .

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Subspaces

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Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following three conditions:
The zero vector, 0, is in W.
If u and v are elements of W, then any linear combination of u and v is an element of W;
If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;

Front 3

Linear combination

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Suppose that K is a field (for example, the real numbers) and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars. If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is

Front 4

SYSTEMS OF EQUATIONS

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A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system.


The equations in the system can be linear or non-linear.

Front 5

Change of coordinate matrix

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Let β and β
0
be two ordered bases for a nite-dimensional vector
space V , and let Q = [IV ]
β
β
0 . Then
1 Q is invertible.
2 For any v ∈ V , [v]β = Q[v]
β

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Dimension theorem for vector spaces

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Given a vector space V, any two linearly independent generating sets (in other words, any two bases) have the same cardinality.