Emma114's Notes

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Course notes on matrix algebra, determinants, systems of linear equations, subspaces, and introductory study of eigenvalues and eigenvectors

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Week 1: Systems of Linear Equations, Row Operations, and Gaussian Elimination • Class notes
Week 2: Rank, Homogeneous Systems, and Matrix Algebra • Class notes
Week 3: Matrix Inverse, Elementary Matrices, and Inverse Properties • Class notes
Week 4: Determinants, Spanning Sets, and Linear Independence • Class notes
Week 5: Subspace, Basis, and Dimension, Row Column, Null Space, Rank-Nullity Theorem • Class notes
And more ….

Week 10: Applications of Eigenvalues and Eigenvectors

Uses of eigenvalues, particularly in dynamic systems. Revisits Markov chains. Using matrix powers, the long-term behavior of the system is analyzed through the steady-state vector.

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Week 9: Applications of Eigenvalues and Eigenvectors

Focuses on practical uses of eigenvalues and eigenvectors, particularly in modeling systems over time. Diagonalization and introduces Markov chains, using transition matrices to model population. Discusses concept of a steady-state vector representing long-term equilibrium.

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• 3 pages •
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Week 8: Eigenspaces and Diagonalization & Complex Eigenvalues

Expands on diagonalization and introduces complex eigenvalues. It reviews conditions under which a matrix is diagonalizable. Discusses the Fundamental Theorem of Algebra. Defines complex numbers, their conjugates, and modulus, and explains complex conjugate pairs.

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Week 7: Eigenvalues and Eigenvectors

Introduces the concepts of eigenvalues and eigenvectors, how a matrix transforms vectors by stretching or reversing their direction. Explains how that eigenvectors are found. Also covers diagonalization. It includes conditions for diagonalizability and shows how the eigenvalues of a diagonal matrix are just its diagonal entries.

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Week 6: Linear Transformations, Matrices, Kernel and Image of a Linear Transformation

Focuses on expressing linear transformations as matrix operations. Covers how to find matrices for transformations like rotations and reflections. Includes the kernel and the image. Explains how to compute transformation matrices and find bases for the kernel and image through row reduction.

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Week 5: Subspace, Basis, and Dimension, Row Column, Null Space, Rank-Nullity Theorem

Introduces subspaces, bases, dimension, null space, column space, and row space. Explains how to find bases using row reduction, defines rank and nullity, and presents the Rank-Nullity Theorem. Shows how to determine if vectors form a basis and interpret matrices as linear transformations.

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Week 4: Determinants, Spanning Sets, and Linear Independence

Lecture notes on determinants, spanning sets, and linear independence. Properties and calculations of determinants, including the use of cofactor expansion (Laplace expansion), properties under row operations, and special results. The notes also explain how determinants behave under scalar multiplication and row swaps. Defines the span of a set of vectors algebraicall. Linear independence, examples to determine whether sets of vectors are linearly independent.

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• 4 pages •
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MTH 341 • MTH 341
Linear Algebra Course Notes • By emma114

Week 3: Matrix Inverse, Elementary Matrices, and Inverse Properties

Introduces elementary matrices, row swapping, and row addition. Explains how to construct a product of elementary matrices to represent a sequence of row operations. Discusses matrix inverses, including how to compute them. Invertibility of elementary matrices, the behavior of inverses under transposition and multiplication, and conditions for a matrix to be invertible based on its row-reduced echelon form (RREF).

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Week 2: Rank, Homogeneous Systems, and Matrix Algebra

Lecture notes on the concept of homogeneous systems; difference between trivial and nontrivial solutions. The notes define matrix rank and connects this to the number of free variables in a solution set. Covers the idea of expressing solutions as linear combinations of basic solutions. Matrix algebra, scalar multiplication, matrix multiplication, and transposition, along with their associated properties. Introduces symmetry and skew-symmetry in matrices. Defines the matrix inverse.

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Class notes
• 4 pages •
by emma114 •
uploaded 2025