Linear Algebra and Geometry 2 For Free

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Ruchika oberoi

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Mar 27, 2022
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[Download] Linear Algebra and Geometry 2 For Free

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What you’ll learn

  • How to solve problems in linear algebra and geometry (illustrated with 153 solved problems) and why these methods work.
  • Important concepts concerning vector spaces, such as basis, dimension, coordinates, and subspaces.
  • Linear combinations, linear dependence and independence in various vector spaces, and how to interpret them geometrically in R2 and R3.
  • How to recalculate coordinates from one basis to another, both with help of transition matrices and by solving systems of equations.
  • Row space, columns space and nullspace for matrices, and about usage of these concepts for solving various types of problems.
  • Linear transformations: different ways of looking at them (as matrix transformations, as transformations preserving linear combinations).
  • How to compose linear transformations and how to compute their standard matrices in different bases; compute the kernel and the image for transformations.
  • Understand the connection between matrices and linear transformations, and see various concepts in accordance with this connection.
  • Work with various geometrical transformations in R2 and R3, be able to compute their matrices and explain how these transformations work.
  • Understand the concept of isometry and be able to give some examples, and formulate their connection with orthogonal matrices.
  • Transform any given basis for a subspace of Rn into an orthonormal basis of the same subspace with help of Gram-Schmidt Process.
  • Compute eigenvalues, eigenvectors, and eigenspaces for a given matrix, and give geometrical interpretations of these concepts.
  • Determine whether a given matrix is diagonalizable or not, and perform its diagonalization if it is.
  • Understand the relationship between diagonalizability and dimensions of eigenspaces for a matrix.
  • Use diagonalization for problem solving involving computing the powers of square matrices, and motivate why this method works.
  • Be able to formulate and use The Invertible Matrix Theorem and recognise the situations which are suitable for the determinant test (and which are not).
  • Use Wronskian to determine whether a set of smooth functions is linearly independent or not; be able to compute Vandermonde determinant.
  • Work with various vector spaces, for example with Rn, the space of all n-by-m matrices, the space of polynomials, the space of smooth functions.

Requirements

  • Linear Algebra and Geometry 1 (systems of equations, matrices and determinants, vectors and their products, analytic geometry of lines and planes)
  • High-school and college mathematics (mainly arithmetic, some trigonometry, polynomials)
  • Some basic calculus (used in some examples)
  • Basic knowledge of complex numbers (used in an example)

Who this course is for:

  • University and college engineering



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