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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/02%3A_Matrix_Arithmetic/2.04%3A_Vector_Solutions_to_Linear_Systems\[\begin{align}\begin{aligned}\vec{x}&=\left[\begin{array}{c}{x_{1}}\\{x_{2}}\\{x_{3}}\\{x_{4}}\\{x_{5}}\end{array}\right] \\ &=\left[\begin{array}{c}{-8-11x_{3}+2x_{4}+11x_{5}}\\{5+7x_{3}-2x_{4}-9x_{...\[\begin{align}\begin{aligned}\vec{x}&=\left[\begin{array}{c}{x_{1}}\\{x_{2}}\\{x_{3}}\\{x_{4}}\\{x_{5}}\end{array}\right] \\ &=\left[\begin{array}{c}{-8-11x_{3}+2x_{4}+11x_{5}}\\{5+7x_{3}-2x_{4}-9x_{5}}\\{x_{3}}\\{x_{4}}\\{x_{5}}\end{array}\right] \\ &=\left[\begin{array}{c}{-8}\\{5}\\{0}\\{0}\\{0}\end{array}\right]+\left[\begin{array}{c}{-11x_{3}}\\{7x_{3}}\\{x_{3}}\\{0}\\{0}\end{array}\right]+\left[\begin{array}{c}{2x_{4}}\\{-2x_{4}}\\{0}\\{x_{4}}\\{0}\end{array}\right]+\left[\begin{array}{c…
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/01%3A_Systems_of_Linear_Equations/1.03%3A_Elementary_Row_Operations_and_Gaussian_EliminationMost of the time\(^{1}\) we will want to take our original matrix and, using the elementary row operations, put it into something called reduced row echelon form.\(^{2}\) This is our “destination,” fo...Most of the time\(^{1}\) we will want to take our original matrix and, using the elementary row operations, put it into something called reduced row echelon form.\(^{2}\) This is our “destination,” for this form allows us to readily identify whether or not a solution exists, and in the case that it does, what that solution is.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/02%3A_Matrix_Arithmetic/2.02%3A_Matrix_Multiplication\[\begin{aligned}AB+AC&=\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right]\left[\begin{array}{cc}{1}&{1}\\{1}&{-1}\end{array}\right]+\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right]\le...\[\begin{aligned}AB+AC&=\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right]\left[\begin{array}{cc}{1}&{1}\\{1}&{-1}\end{array}\right]+\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right]\left[\begin{array}{cc}{2}&{1}\\{1}&{2}\end{array}\right] \\ &=\left[\begin{array}{cc}{3}&{-1}\\{7}&{-1}\end{array}\right]+\left[\begin{array}{cc}{4}&{5}\\{10}&{11}\end{array}\right] \\ &=\left[\begin{array}{cc}{7}&{4}\\{17}&{10}\end{array}\right]\end{aligned} \nonumber \]
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/04%3A_Eigenvalues_and_Eigenvectors/4.01%3A_Eigenvalues_and_Eigenvectors/4.1.1%3A_Exercises_4.1\(A=\left[\begin{array}{ccc}{-11}&{-19}&{14}\\{-6}&{-8}&{6}\\{-12}&{-22}&{15}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{3}\\{2}\\{4}\end{array}\right]\) \(A=\left[\begin{array}{ccc}{-7}&{1}...\(A=\left[\begin{array}{ccc}{-11}&{-19}&{14}\\{-6}&{-8}&{6}\\{-12}&{-22}&{15}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{3}\\{2}\\{4}\end{array}\right]\) \(A=\left[\begin{array}{ccc}{-7}&{1}&{3}\\{10}&{2}&{-3}\\{-20}&{-14}&{1}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{1}\\{-2}\\{4}\end{array}\right]\) \(A=\left[\begin{array}{ccc}{-12}&{-10}&{0}\\{15}&{13}&{0}\\{15}&{18}&{-5}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]\)
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/01%3A_Systems_of_Linear_Equations/1.04%3A_Existence_and_Uniqueness_of_SolutionsIn the “or not” case, the constants determine whether or not infinite solutions or no solution exists. (So if a given linear system has exactly one solution, it will always have exactly one solution e...In the “or not” case, the constants determine whether or not infinite solutions or no solution exists. (So if a given linear system has exactly one solution, it will always have exactly one solution even if the constants are changed.) Let’s look at an example to get an idea of how the values of constants and coefficients work together to determine the solution type.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/02%3A_Matrix_Arithmetic/2.03%3A_Visualizing_Matrix_Arithmetic_in_2DNow that we know how to compute the length of a vector, let’s revisit a statement we made as we explored Examples \(\PageIndex{4}\) and \(\PageIndex{5}\): “Multiplying a vector by a positive scalar \(...Now that we know how to compute the length of a vector, let’s revisit a statement we made as we explored Examples \(\PageIndex{4}\) and \(\PageIndex{5}\): “Multiplying a vector by a positive scalar \(c\) stretches the vectors by a factor of \(c\) \(\ldots\)” At that time, we did not know how to measure the length of a vector, so our statement was unfounded.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)This text deals with matrix algebra, as opposed to linear algebra. Without arguing semantics, I view matrix algebra as a subset of linear algebra, focused primarily on basic concepts and solution tech...This text deals with matrix algebra, as opposed to linear algebra. Without arguing semantics, I view matrix algebra as a subset of linear algebra, focused primarily on basic concepts and solution techniques. There is little formal development of theory and abstract concepts are avoided. This is akin to the master carpenter teaching his apprentice how to use a hammer, saw and plane before teaching how to make a cabinet.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/04%3A_Eigenvalues_and_Eigenvectors/4.02%3A_Properties_of_Eigenvalues_and_EigenvectorsIn this section we’ll explore how the eigenvalues and eigenvectors of a matrix relate to other properties of that matrix. This section is essentially a hodgepodge of interesting facts about eigenvalue...In this section we’ll explore how the eigenvalues and eigenvectors of a matrix relate to other properties of that matrix. This section is essentially a hodgepodge of interesting facts about eigenvalues; the goal here is not to memorize various facts about matrix algebra, but to again be amazed at the many connections between mathematical concepts.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/03%3A_Operations_on_MatricesIn the previous chapter we learned about matrix arithmetic: adding, subtracting, and multiplying matrices, finding inverses, and multiplying by scalars. In this chapter we learn about some operations ...In the previous chapter we learned about matrix arithmetic: adding, subtracting, and multiplying matrices, finding inverses, and multiplying by scalars. In this chapter we learn about some operations that we perform on matrices. We can think of them as functions: you input a matrix, and you get something back. With the other operations, the trace and the determinant, we input matrices and get numbers in return, an idea that is different than what we have seen before.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/05%3A_Graphical_Explorations_of_Vectors/5.02%3A_Properties_of_Linear_Transformations\[T_{1}\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+1}\\{x_{2}}\end{array}\right]\qquad T_{2}\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{arra...\[T_{1}\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+1}\\{x_{2}}\end{array}\right]\qquad T_{2}\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}/x_{2}}\\{\sqrt{x_{2}}}\end{array}\right] \qquad T_{3}\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{\sqrt{7}x_{1}-x_{2}}\\{\pi x_{2}}\end{array}\right]\nonumber \]
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/11%3A_Appendices/06%3A_Eigenvalues_and_Eigenvectors\[\begin{align}\begin{aligned}\left[\begin{array}{cc}{1}&{4}\\{2}&{3}\end{array}\right]\left[\begin{array}{c}{2}\\{2}\end{array}\right]&=\left[\begin{array}{c}{10}\\{10}\end{array}\right]=5\left[\begi...\[\begin{align}\begin{aligned}\left[\begin{array}{cc}{1}&{4}\\{2}&{3}\end{array}\right]\left[\begin{array}{c}{2}\\{2}\end{array}\right]&=\left[\begin{array}{c}{10}\\{10}\end{array}\right]=5\left[\begin{array}{c}{2}\\{2}\end{array}\right]; \\ \left[\begin{array}{cc}{1}&{4}\\{2}&{3}\end{array}\right]\left[\begin{array}{c}{7}\\{7}\end{array}\right]&=\left[\begin{array}{c}{35}\\{35}\end{array}\right]=5\left[\begin{array}{c}{7}\\{7}\end{array}\right]; \\ \left[\begin{array}{cc}{1}&{4}\\{2}&{3}\end{a…
