11 Elements of the general theory of the linear ODE
... Q: Can you show that operator dt 2 is linear? Now we can give a general definition of a linear equation, which works equally well for algebraic equations, matrix equations, differential equations, integral equations (can you think of an example?), ...
... Q: Can you show that operator dt 2 is linear? Now we can give a general definition of a linear equation, which works equally well for algebraic equations, matrix equations, differential equations, integral equations (can you think of an example?), ...
Ch 4.3 Quick Graphs Using Intercepts
... • In this lesson we talked about Quick graphs using intercepts. Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… • I will give you credit for doing this lesson…please see the next slide… ...
... • In this lesson we talked about Quick graphs using intercepts. Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… • I will give you credit for doing this lesson…please see the next slide… ...
1 Lecture 10: Math 285 (Bronski) Existence
... is y = cos(x), which is not a linear combination of sin(x) and 2 sin(x). So the important thing is to distinguish when two solutions are really “different”. This is not always so clear, especially in the case of higher order equations. We’ll start with a simple definition, which will need to be modi ...
... is y = cos(x), which is not a linear combination of sin(x) and 2 sin(x). So the important thing is to distinguish when two solutions are really “different”. This is not always so clear, especially in the case of higher order equations. We’ll start with a simple definition, which will need to be modi ...
Lecture 1. Modules
... r(m + N ) = rm + N for all r ∈ R, m ∈ M. Definition. If M and N are R-modules, a mapping ϕ : M → N is called a homomorphism of R-modules (alternatively ϕ is an R-linear mapping) if (1) ϕ is a homomorphism of abelian group (2) ϕ(rm) = rϕ(m) for all r ∈ R, m ∈ M . 1.4. Modules and group actions. Defin ...
... r(m + N ) = rm + N for all r ∈ R, m ∈ M. Definition. If M and N are R-modules, a mapping ϕ : M → N is called a homomorphism of R-modules (alternatively ϕ is an R-linear mapping) if (1) ϕ is a homomorphism of abelian group (2) ϕ(rm) = rϕ(m) for all r ∈ R, m ∈ M . 1.4. Modules and group actions. Defin ...
CORE 4 Summary Notes
... real number and is sometimes called a scalar multiple of a. - a has the same magnitude but is in the opposite direction to a. ...
... real number and is sometimes called a scalar multiple of a. - a has the same magnitude but is in the opposite direction to a. ...
ch2_1lecture
... • We ‘ resolve’ vectors into components using the x and y axes system • Each component of the vector is shown as a magnitude and a direction. • The directions are based on the x and y axes. We use the “unit vectors” i and j to designate the x and y axes. ...
... • We ‘ resolve’ vectors into components using the x and y axes system • Each component of the vector is shown as a magnitude and a direction. • The directions are based on the x and y axes. We use the “unit vectors” i and j to designate the x and y axes. ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.