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Appendix A: Linear Algebra

Last updated

Feb 24, 2025
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- 8.E: Nonlinear Equations (Exercises)
- A.1: Vectors, Mappings, and Matrices

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A.1: Vectors, Mappings, and Matrices
The page discusses the importance of linear algebra in solving engineering problems by using finite-dimensional linear problems, which are easier to handle with matrix manipulation. It explains the concept of vectors in $n$ -dimensional space ${\mathbb R}^n$ , operations on vectors like addition, subtraction, and scaling, and introduces the notion of linear mappings and matrices.
A.2: Matrix Algebra
This page provides a comprehensive overview of matrices, covering one-by-one matrices, matrix addition, scalar multiplication, matrix multiplication, and properties of matrices. It explains the operations related to matrices such as adding and multiplying matrices and scalars, including special cases like identity matrices and inverse matrices. The concept of transpose is also introduced, with examples demonstrating how swapping rows and columns affects matrix multiplication.
A.3: Elimination
This page provides an in-depth explanation of solving linear systems of equations using matrices and techniques such as elimination and Gauss-Jordan elimination. It discusses transforming systems into matrix equations, performing row operations to achieve row echelon or reduced row echelon form, and distinguishing cases where solutions are unique, non-unique, or non-existent. Concepts like linear independence, rank, nullspace, and kernel are introduced alongside practical examples.
A.4: Subspaces, Dimension, and The Kernel
The page discusses concepts related to subspaces, basis, and dimension in linear algebra. It explores the structure of the solutions to a linear equation $L\vec{x} = \vec{0}$ , identifying them as a subspace of ${\mathbb R}^n$ , where operations like addition and scalar multiplication remain within the subspace. A basis for a subspace consists of linearly independent vectors that span the subspace, defining its dimension.
A.5: Inner Product and Projections
The page discusses the concepts of inner product and orthogonality in vector spaces, particularly in ${\mathbb{R}}^n$ . It defines the standard inner product as the dot product and explores how it is used to calculate the length of vectors and angles between them. Orthogonal vectors, which form right angles, have an inner product of zero. The text introduces the concept of orthogonal and orthonormal bases, explaining how vectors can be expressed in terms of these bases.
A.6: Determinant
The page explains the concept of the determinant of square matrices, including definitions for $1 \times 1$ and $2 \times 2$ matrices. It elaborates on how determinants map $n$ -dimensional spaces and change the area of objects, explaining the determinant as a factor indicating changes in area and orientation. The cofactor expansion is introduced for computing determinants of larger matrices.
A.E: Linear Algebra (Exercises)
This page contains a series of linear algebra exercises, covering topics such as vectors, matrices, linear mappings, and determinants. It provides practice problems on drawing vectors, computing magnitudes, matrix operations, row reduction, and finding bases for subspaces. It also includes applications of the Gram-Schmidt process for orthogonalization and explores inner product spaces along with projections.

Thumbnail: 3 planes intersect at a point. (CC BY-SA 4.0; Fred the Oyster).

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