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- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.02%3A_The_Inverse_of_a_MatrixThis page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row...This page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row operations. It provides examples illustrating both the derivation of inverses and scenarios where matrices lack inverses.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_DeterminantsThere are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of...There are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of row operations on the determinant of a matrix. In future sections, we will see that using the following properties can greatly assist in finding determinants. This section will use the theorems as motivation to provide various examples of the usefulness of the properties.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.01%3A_Matrix_OperationsYou have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that matrices are important not only for s...You have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that matrices are important not only for systems of equations but also in many applications. In this section, we explore some matrix operations.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.02%3A_The_Method_of_Undetermined_Coefficients_IThis section present the method of undetermined coefficients, which can be used to solve nonhomogeneous equations of the form ay''+by'+cy=F(x) where a, b, and c are constants and F(x) has a special ...This section present the method of undetermined coefficients, which can be used to solve nonhomogeneous equations of the form ay''+by'+cy=F(x) where a, b, and c are constants and F(x) has a special form that is still sufficiently general to occur in many applications. This sections makes extensive use of the idea of variation of parameters introduced previously.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.02%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.04%3A_Matrix_MultiplicationThis page explores the interplay between compositions of transformations and matrix multiplication in linear algebra. It defines the composition of transformations, illustrates their properties, inclu...This page explores the interplay between compositions of transformations and matrix multiplication in linear algebra. It defines the composition of transformations, illustrates their properties, including non-commutativity and associativity, and connects these concepts to matrix operations. The Row-Column Rule for matrix multiplication is explained, alongside the implications of this non-commutative nature.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.05%3A_The_Chain_Rule_for_Multivariable_FunctionsIn single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is...In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. In this section, we study extensions of the chain rule and learn how to take derivatives of compositions of functions of more than one variable.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/05%3A_Graph_Theory/5.01%3A_Discovering_GraphsThis page explains how to define graphs through various properties, using examples and checkpoints to distinguish valid graph formations from invalid ones. It employs set notation and prompts readers ...This page explains how to define graphs through various properties, using examples and checkpoints to distinguish valid graph formations from invalid ones. It employs set notation and prompts readers to analyze necessary and forbidden properties of graphs, facilitating a translation between set forms and diagrams.
- https://math.libretexts.org/Courses/Stanford_Online_High_School/Logic_for_All%3A_An_Introduction_to_Logical_Reasoning/07%3A_The_BiconditionalThis page covers biconditionals, defined by "if and only if," indicating that two statements hold the same truth value. It includes explanations, examples, and exercises to practice identifying and re...This page covers biconditionals, defined by "if and only if," indicating that two statements hold the same truth value. It includes explanations, examples, and exercises to practice identifying and rewriting biconditionals while emphasizing their significance in logic, mathematics, and computer science. The text provides truth tables, real-life scenarios, and puzzles to illustrate their dual implications.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.02%3A_The_Matrix_of_a_Linear_Transformation_IIn the previous section we saw that multiplication by a matrix is a linear transformation. It turns out that it is always the case that if T is any linear transformation which maps \(\mathbb{R}^{n...In the previous section we saw that multiplication by a matrix is a linear transformation. It turns out that it is always the case that if T is any linear transformation which maps Rn to Rm, then it corresponds to multiplication by a matrix.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.06%3A_BasisIn this section we extend a linearly independent set and shrink a spanning set to a basis of a given vector space.
