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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/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.04%3A_Linear_IndependenceIn this section, we will again explore concepts introduced earlier in terms of Rn and extend them to apply to abstract vector spaces.
- 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/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.11%3A_The_Matrix_of_a_Linear_TransformationYou may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary v...You may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another.
- 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/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.08%3A_Linear_Transformations/7.8E%3A_Exercises_for_Section_7.8This page contains exercises on linear transformations from polynomial spaces to real numbers or vectors, focusing on determining transformations for various polynomial degrees. It emphasizes the use ...This page contains exercises on linear transformations from polynomial spaces to real numbers or vectors, focusing on determining transformations for various polynomial degrees. It emphasizes the use of linearity properties and derives results from known transformations. Exercise 6 examines functions that fail to meet linearity criteria, while the final exercise proves the linear independence of vectors through the independence of their transformations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.01%3A_Linear_TransformationsIn this section we will consider how matrix multiplication transforms vectors while preserving vector addition and scalar multiplication. Such transformations are called linear transformations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.09%3A_Isomorphisms/7.9E%3A_Exercises_for_Section_7.9This page discusses exercises on linear transformations, highlighting that a transformation is an isomorphism if and only if its matrix is invertible. It emphasizes the significance of dimensions for ...This page discusses exercises on linear transformations, highlighting that a transformation is an isomorphism if and only if its matrix is invertible. It emphasizes the significance of dimensions for injectivity and surjectivity. The exercises encourage analyzing specific transformations and their properties, as well as combinations of transformations.
