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Felix Quintana
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I am self teaching the subject but I am unsure of what is the whole point and picture
Hey, I just started going over these concepts!, not kernel yet, but thanks! This was very helpful.mathwonk said:well that was really an excellent answer to the sentence in your post, i.e. essentially what is linear algebra good for. your title question however seemed to me more of what in linear algebra should you learn, rather than why. so i was going to say, more narrowly, I suggest you should learn the concept of linearity and its consequences for solving (linear) differential equarions. foremost learn the concept of linear dimension of vector spaces, and how it allows you to classify all vector spaces. Starting on linear maps, learn the key concept of the "kernel" of a map and its use in computing general solutions and rank of a map. Then I recommend studying closely the solutions of linear constant coefficient differential equations, as this gives you the model for classifying all linear mappings in finite dimensions. I.e. over the algebraic closure of your scalar field at least, every linear map in finite dimensions looks in some basis like (a sum of copies of) the derivative map acting on the solution set of a linear ode. This is called "Jordan form". Then for applications to Fourier series and geometry, you should learn something about linear maps in the presence of the notion of dot product angle and length, as well as some more abstract forms of "duality", i.e. study especially scalar valued linear functions acting on your vector space. Finally for computations, learn about characteristic polynomials, determinants, and Gaussian reduction of matrices.
In a nutshell, the concept of dimension is fundamental because two vector spaces are essentially the same, i.e. "isomorphic", if and only if they have the same dimension. Moreover two linear maps between the same two (different) vector spaces are the same up to isomorphisms of source and target if and only if their kernels have the same dimension. And two linear maps S,T of the same vector space to itself are the same up to isomorphism of that space, if and only if they have the same characteristic polynomial, and the dimensions of the kernels of all the powers (S-c)^r, (T-c)^r have kernels of the same dimension, where c is any root of the common characteristic polynomial.
Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It involves the study of linear transformations and their properties, as well as systems of linear equations and their solutions.
Linear algebra is a fundamental tool in many branches of science, including physics, engineering, and computer science. It is used to model and solve complex systems, from predicting the movement of objects in space to analyzing data in machine learning algorithms.
Some key concepts in linear algebra include vectors, matrices, systems of linear equations, determinants, eigenvalues, and eigenvectors. These concepts are used to represent and manipulate linear equations and transformations.
Linear algebra has numerous real-life applications, such as in computer graphics, data analysis, and optimization problems. It is also used in fields such as economics, biology, and social sciences to model and analyze complex systems.
To learn linear algebra, one should have a strong foundation in algebra and basic mathematical operations. It also requires critical thinking and problem-solving skills, as well as the ability to visualize and manipulate mathematical concepts in a geometric or abstract way.