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1: Functions and Graphs
https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/01%3A_Functions_and_Graphs
In this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functio...In this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functions, and we show the properties of their graphs. We provide examples of equations with terms involving these functions and illustrate the algebraic techniques necessary to solve them. In short, this chapter provides the foundation for the material to come.
1: Systems of Linear Equations- Algebra
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/01%3A_Systems_of_Linear_Equations-_Algebra
This page discusses the algebraic study of linear equations, detailing methods for solving them, particularly through row reduction. It explains a systematic approach to solving equations and how to e...This page discusses the algebraic study of linear equations, detailing methods for solving them, particularly through row reduction. It explains a systematic approach to solving equations and how to express solutions in parametric form. The content is organized into sections that build foundational knowledge on linear equations, algorithms for solutions, and solution representation.
2.8E: Exercises for Section 2.8
https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.08%3A_Exponential_Growth_and_Decay/2.8E%3A_Exercises_for_Section_2.8
This page features true or false exercises focused on exponential functions, including growth and decay, along with financial calculations. It offers solutions and explanations for problems related to...This page features true or false exercises focused on exponential functions, including growth and decay, along with financial calculations. It offers solutions and explanations for problems related to population growth models, radioactive decay, interest rates, and more. Specific topics include calculating doubling times, analyzing population changes, and understanding temperature dynamics.
3.3: Trigonometric Substitution
https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.03%3A_Trigonometric_Substitution
The technique of trigonometric substitution comes in very handy when evaluating integrals of certain forms. This technique uses substitution to rewrite these integrals as trigonometric integrals.
7.4: Linear Independence
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.04%3A_Linear_Independence
In this section, we will again explore concepts introduced earlier in terms of Rn and extend them to apply to abstract vector spaces.
5.3: Similarity
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.3%3A_Similarity
This page explores similar matrices defined by the relation $A = CBC^{-1}$ , focusing on their geometric interpretations, eigenvalues, eigenvectors, and properties as an equivalence relation. It expl...This page explores similar matrices defined by the relation $A = CBC^{-1}$ , focusing on their geometric interpretations, eigenvalues, eigenvectors, and properties as an equivalence relation. It explains how to compute matrix powers, emphasizing transformations and changes of coordinates between different systems. The relationship between matrices $A$ and $B$ is examined, highlighting how they share characteristics like trace and determinant but may differ in eigenvectors.
1.7: Existence and Uniqueness of Solutions of Nonlinear Equations
https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.07%3A_Existence_and_Uniqueness_of_Solutions_of_Nonlinear_Equations
Although there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximati...Although there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximations, it is useful to know conditions that imply the existence and uniqueness of solutions of initial value problems for nonlinear equations. In this section we state such a condition and illustrate it with examples.
3.8: Implicit Differentiation
https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/03%3A_Derivatives/3.08%3A_Implicit_Differentiation
We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations). By using implicit differentiation, we can find the equation of a tangent line to t...We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations). By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.
4.4: Spanning Sets in Rⁿ
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.04%3A_Spanning_Sets_in_R
By generating all linear combinations of a set of vectors one can obtain various subsets of $\mathbb{R}^{n}$ which we call subspaces. For example what set of vectors in $\mathbb{R}^{3}$ generate t...By generating all linear combinations of a set of vectors one can obtain various subsets of $\mathbb{R}^{n}$ which we call subspaces. For example what set of vectors in $\mathbb{R}^{3}$ generate the $XY$ -plane? What is the smallest such set of vectors can you find? The tools of spanning, linear independence and basis are exactly what is needed to answer these and similar questions and are the focus of this section.
6.10: Quadratic Forms
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.10%3A_Quadratic_Forms
In this section we use the techniques learned in this chapter to investigate quadratic forms.
7.3E: Exercises for Section 7.3
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.03%3A_Spanning_Sets/7.3E%3A_Exercises_for_Section_7.3
This page discusses exercises focused on spans of vectors and matrices. It confirms that the zero vector is always included in any span, determines the inclusion of polynomials in specific spans, and ...This page discusses exercises focused on spans of vectors and matrices. It confirms that the zero vector is always included in any span, determines the inclusion of polynomials in specific spans, and verifies a matrix's inclusion in a span. Lastly, it requires proving that a set spans all 2x2 matrices.

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