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- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.07%3A_The_Dot_ProductThere are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product. When we take the dot product of vectors, the result is a scalar. ...There are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product. When we take the dot product of vectors, the result is a scalar. For this reason, the dot product is also called the scalar product and sometimes the inner product.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/10%3A_Some_Prerequisite_Topics/10.02%3A_Well_Ordering_and_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.11%3A_OrthogonalityIn this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.09%3A_The_Cross_ProductRecall that the dot product is one of two important products for vectors. The second type of product for vectors is called the cross product.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/09%3A_Vector_Spaces/9.06%3A_Linear_Transformations\[\begin{aligned} x^2 & = \textstyle \frac{1}{2}(x^2+x) + \frac{1}{2}(x^2-x) \\ x & = \textstyle \frac{1}{2}(x^2+x) - \frac{1}{2}(x^2-x) \\ 1 & = (x^2+1)-\textstyle \frac{1}{2}(x^2+x) - \frac{1}{2}(x^...x2=12(x2+x)+12(x2−x)x=12(x2+x)−12(x2−x)1=(x2+1)−12(x2+x)−12(x2−x). Then \[\begin{aligned} T(x^2) & = \textstyle T\left(\frac{1}{2}(x^2+x) + \frac{1}{2}(x^2-x)\right) =\frac{1}{2}T(x^2+x) + \frac{1}{2}T(x^2-x)\\ & = \textstyle \frac{1}{2}(-1) + \frac{1}{2}(1) = 0. \\ T(x) & = \textstyle T\left(\frac{1}{2}(x^2+x) - \frac{1}{2}(x^2-x)\right) = \f…
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.05%3A_Geometric_Meaning_of_Scalar_MultiplicationThen, by using Definition 4.4.1, the length of this vector is given by \[\sqrt{\left( \left( k u_{1}\right) ^{2}+\left( k u_{2}\right) ^{2}+\left( k u_{3}\right) ^{2}\right) }=\left\vert k \right\vert...Then, by using Definition 4.4.1, the length of this vector is given by √((ku1)2+(ku2)2+(ku3)2)=|k|√u21+u22+u23 Thus the following holds. ‖ In other words, multiplication by a scalar magnifies or shrinks the length of the vector by a factor of \left\vert k \right\vert.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/05%3A_Linear_Transformations/5.05%3A_One-to-One_and_Onto_TransformationsThis section is devoted to studying two important characterizations of linear transformations, called One to One and Onto.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/09%3A_Vector_Spaces
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.08%3A_Planes_in_RMuch like the above discussion with lines, vectors can be used to determine planes in \mathbb{R}^n.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/05%3A_Linear_Transformations/5.07%3A_The_Kernel_and_Image_of_A_Linear_MapIn this section we will consider the case where the linear transformation is not necessarily an isomorphism.
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/11%3A_Systems_of_Equations/11.02%3A_Elementary_OperationsWe have taken an in depth look at graphical representations of systems of equations, as well as how to find possible solutions graphically. Our attention now turns to working with systems algebraicall...We have taken an in depth look at graphical representations of systems of equations, as well as how to find possible solutions graphically. Our attention now turns to working with systems algebraically.
