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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/05%3A_Vector_SpacesThe two key properties of vectors are that they can be added together and multiplied by scalars, so we make the following definition.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/05%3A_Vector_Spaces/5.01%3A_Examples_of_Vector_SpacesOne can find many interesting vector spaces, such as the following:
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/03%3A_Vector_Spaces_and_Metric_Spaces/3.05%3A_Vector_Spaces._The_Space_C._Euclidean_Spaces\[\begin{aligned}\left|t x+y^{\prime}\right|^{2} &=\left(t x+y^{\prime}\right) \cdot\left(t x+y^{\prime}\right) \\ &=t x \cdot t x+y^{\prime} \cdot t x+t x \cdot y^{\prime}+y^{\prime} \cdot y^{\prime}...|tx+y′|2=(tx+y′)⋅(tx+y′)=tx⋅tx+y′⋅tx+tx⋅y′+y′⋅y′=t2(x⋅x)+t(y′⋅x)+t(x⋅y′)+(y′⋅y′)
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.03%3A_Linear_TransformationsThis page covers linear transformations and their connections to matrix transformations, defining properties necessary for linearity and providing examples of both linear and non-linear transformation...This page covers linear transformations and their connections to matrix transformations, defining properties necessary for linearity and providing examples of both linear and non-linear transformations. It highlights the importance of the zero vector, standard coordinate vectors, and defines transformations like rotations, dilations, and the identity transformation.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/20%3A_Vector_SpacesIn a physical system a quantity can often be described with a single number. For example, we need to know only a single number to describe temperature, mass, or volume. To give the location of a point...In a physical system a quantity can often be described with a single number. For example, we need to know only a single number to describe temperature, mass, or volume. To give the location of a point in space, we need x, y, and z coordinates. Temperature distribution over a solid object requires four numbers: three to identify each point within the object and a fourth to describe the temperature at that point.
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/11%3A_Vector_SpacesThe two key properties of vectors are that they can be added together and multiplied by scalars, so we make the following definition.
- https://math.libretexts.org/Under_Construction/Purgatory/Remixer_University/Username%3A_junalyn2020/Book%3A_Introduction_to_Real_Analysis_(Lebl)/8%3A_Several_Variables_and_Partial_Derivatives/8.1%3A_Vector_Spaces%2C_linear_Mappings%2C_and_ConvexityLet X be a set together with operations of addition, +:X×X→X, and multiplication, ⋅:R×X→X, (we write ax instead of a⋅x). \(X...Let X be a set together with operations of addition, +:X×X→X, and multiplication, ⋅:R×X→X, (we write ax instead of a⋅x). X is called a vector space (or a real vector space) if the following conditions are satisfied: (Addition is associative) If u,v,w∈X, then u+(v+w)=(u+v)+w. (Addition is commutative) If u,v∈X, then u+v=v+u. (Additive identity) There is a 0∈X such that v+0=v…
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/11%3A_Vector_Spaces/11.01%3A_Examples_of_Vector_SpacesOne can find many interesting vector spaces, such as the following:
