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<!DOCTYPE html>
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<title>Vectors and Matrices 2021 - 2022</title>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Lecture 1</a>
<ul>
<li class="chapter" data-level="0.1" data-path="index.html"><a href="index.html#contents"><i class="fa fa-check"></i><b>0.1</b> Contents</a></li>
</ul></li>
<li class="chapter" data-level="1" data-path="complex-numbers.html"><a href="complex-numbers.html"><i class="fa fa-check"></i><b>1</b> Complex Numbers</a>
<ul>
<li class="chapter" data-level="1.1" data-path="complex-numbers.html"><a href="complex-numbers.html#definitions"><i class="fa fa-check"></i><b>1.1</b> Definitions</a></li>
<li class="chapter" data-level="1.2" data-path="complex-numbers.html"><a href="complex-numbers.html#basic-properties-and-consequences"><i class="fa fa-check"></i><b>1.2</b> Basic properties and Consequences</a></li>
<li class="chapter" data-level="1.3" data-path="complex-numbers.html"><a href="complex-numbers.html#exponential-and-trigonometric-functions"><i class="fa fa-check"></i><b>1.3</b> Exponential and Trigonometric Functions</a>
<ul>
<li class="chapter" data-level="1.3.1" data-path="complex-numbers.html"><a href="complex-numbers.html#roots-of-units"><i class="fa fa-check"></i><b>1.3.1</b> Roots of units</a></li>
</ul></li>
<li class="chapter" data-level="1.4" data-path="complex-numbers.html"><a href="complex-numbers.html#transformations-lines-and-circles"><i class="fa fa-check"></i><b>1.4</b> Transformations; lines and circles</a></li>
<li class="chapter" data-level="1.5" data-path="complex-numbers.html"><a href="complex-numbers.html#logarithms-and-complex-powers"><i class="fa fa-check"></i><b>1.5</b> Logarithms and Complex Powers</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html"><i class="fa fa-check"></i><b>2</b> Vectors in 3 Dimensions</a>
<ul>
<li class="chapter" data-level="2.1" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#vector-addition-and-scalar-multiplication"><i class="fa fa-check"></i><b>2.1</b> Vector Addition and Scalar Multiplication</a></li>
<li class="chapter" data-level="2.2" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#scalar-or-dot-product"><i class="fa fa-check"></i><b>2.2</b> Scalar or Dot Product</a></li>
<li class="chapter" data-level="2.3" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#orthonormal-bases-and-components"><i class="fa fa-check"></i><b>2.3</b> Orthonormal Bases and Components</a></li>
<li class="chapter" data-level="2.4" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#vector-or-cross-product"><i class="fa fa-check"></i><b>2.4</b> Vector or Cross Product</a></li>
<li class="chapter" data-level="2.5" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#triple-products"><i class="fa fa-check"></i><b>2.5</b> Triple products</a></li>
<li class="chapter" data-level="2.6" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#lines-planes-and-other-vector-equations"><i class="fa fa-check"></i><b>2.6</b> Lines, Planes and Other Vector Equations</a>
<ul>
<li class="chapter" data-level="2.6.1" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#lines"><i class="fa fa-check"></i><b>2.6.1</b> Lines</a></li>
<li class="chapter" data-level="2.6.2" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#planes"><i class="fa fa-check"></i><b>2.6.2</b> Planes</a></li>
<li class="chapter" data-level="2.6.3" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#other-vector-equations"><i class="fa fa-check"></i><b>2.6.3</b> Other Vector Equations</a></li>
</ul></li>
<li class="chapter" data-level="2.7" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#index-suffix-notation-and-the-summation-convention"><i class="fa fa-check"></i><b>2.7</b> Index (suffix) Notation and the Summation convention</a>
<ul>
<li class="chapter" data-level="2.7.1" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#components-delta-and-epsilon"><i class="fa fa-check"></i><b>2.7.1</b> Components; <span class="math inline">\(\delta\)</span> and <span class="math inline">\(\epsilon\)</span></a></li>
<li class="chapter" data-level="2.7.2" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#summation-convention"><i class="fa fa-check"></i><b>2.7.2</b> Summation Convention</a></li>
<li class="chapter" data-level="2.7.3" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#rules"><i class="fa fa-check"></i><b>2.7.3</b> Rules</a></li>
<li class="chapter" data-level="2.7.4" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#application"><i class="fa fa-check"></i><b>2.7.4</b> Application</a></li>
<li class="chapter" data-level="2.7.5" data-path="vectors-in-3-dimensions.html"><a href="vectors-in-3-dimensions.html#epsilon-epsilon-identities"><i class="fa fa-check"></i><b>2.7.5</b> <span class="math inline">\(\epsilon \epsilon\)</span> identities</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="3" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html"><i class="fa fa-check"></i><b>3</b> Vectors in General; <span class="math inline">\(\mathbb{R}^n\)</span> and <span class="math inline">\(\mathbb{C}^n\)</span></a>
<ul>
<li class="chapter" data-level="3.1" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#vectors-in-mathbbrn"><i class="fa fa-check"></i><b>3.1</b> Vectors in <span class="math inline">\(\mathbb{R}^n\)</span></a>
<ul>
<li class="chapter" data-level="3.1.1" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#Rn"><i class="fa fa-check"></i><b>3.1.1</b> Definitions</a></li>
<li class="chapter" data-level="3.1.2" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#cauchy-schwarz-and-triangle-inequalities"><i class="fa fa-check"></i><b>3.1.2</b> Cauchy-Schwarz and Triangle Inequalities</a></li>
<li class="chapter" data-level="3.1.3" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#comments"><i class="fa fa-check"></i><b>3.1.3</b> Comments</a></li>
</ul></li>
<li class="chapter" data-level="3.2" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#vector-spaces"><i class="fa fa-check"></i><b>3.2</b> Vector Spaces</a>
<ul>
<li class="chapter" data-level="3.2.1" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#axioms-span-subspaces"><i class="fa fa-check"></i><b>3.2.1</b> Axioms; span; subspaces</a></li>
<li class="chapter" data-level="3.2.2" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#linear-dependence-and-independence"><i class="fa fa-check"></i><b>3.2.2</b> Linear Dependence and Independence</a></li>
<li class="chapter" data-level="3.2.3" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#inner-product"><i class="fa fa-check"></i><b>3.2.3</b> Inner product</a></li>
</ul></li>
<li class="chapter" data-level="3.3" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#bases-and-dimension"><i class="fa fa-check"></i><b>3.3</b> Bases and Dimension</a></li>
<li class="chapter" data-level="3.4" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#vectors-in-mathbbcn"><i class="fa fa-check"></i><b>3.4</b> Vectors in <span class="math inline">\(\mathbb{C}^n\)</span></a>
<ul>
<li class="chapter" data-level="3.4.1" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#Cn"><i class="fa fa-check"></i><b>3.4.1</b> Definitions</a></li>
<li class="chapter" data-level="3.4.2" data-path="vectors-in-general-mathbbrn-and-mathbbcn.html"><a href="vectors-in-general-mathbbrn-and-mathbbcn.html#inner-product-1"><i class="fa fa-check"></i><b>3.4.2</b> Inner Product</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="4" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html"><i class="fa fa-check"></i><b>4</b> Matrices and Linear Maps</a>
<ul>
<li class="chapter" data-level="4.1" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#introduction"><i class="fa fa-check"></i><b>4.1</b> Introduction</a>
<ul>
<li class="chapter" data-level="4.1.1" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#definitions-1"><i class="fa fa-check"></i><b>4.1.1</b> Definitions</a></li>
<li class="chapter" data-level="4.1.2" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#rank-and-nullity"><i class="fa fa-check"></i><b>4.1.2</b> Rank and Nullity</a></li>
</ul></li>
<li class="chapter" data-level="4.2" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#geometrical-examples"><i class="fa fa-check"></i><b>4.2</b> Geometrical Examples</a>
<ul>
<li class="chapter" data-level="4.2.1" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#rotations"><i class="fa fa-check"></i><b>4.2.1</b> Rotations</a></li>
<li class="chapter" data-level="4.2.2" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#reflections"><i class="fa fa-check"></i><b>4.2.2</b> Reflections</a></li>
<li class="chapter" data-level="4.2.3" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#dilations"><i class="fa fa-check"></i><b>4.2.3</b> Dilations</a></li>
<li class="chapter" data-level="4.2.4" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#shears"><i class="fa fa-check"></i><b>4.2.4</b> Shears</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#matrices-as-linear-maps-mathbbrn-to-mathbbrm"><i class="fa fa-check"></i><b>4.3</b> Matrices as Linear Maps <span class="math inline">\(\mathbb{R}^n \to \mathbb{R}^m\)</span></a>
<ul>
<li class="chapter" data-level="4.3.1" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#definitions-2"><i class="fa fa-check"></i><b>4.3.1</b> Definitions</a></li>
<li class="chapter" data-level="4.3.2" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#examples"><i class="fa fa-check"></i><b>4.3.2</b> Examples</a></li>
<li class="chapter" data-level="4.3.3" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#isometries-area-and-determinants-in-mathbbr2"><i class="fa fa-check"></i><b>4.3.3</b> Isometries, area and determinants in <span class="math inline">\(\mathbb{R}^2\)</span></a></li>
</ul></li>
<li class="chapter" data-level="4.4" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#matrices-for-linear-maps-in-general"><i class="fa fa-check"></i><b>4.4</b> Matrices for Linear Maps in General</a></li>
<li class="chapter" data-level="4.5" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#matrix-algebra"><i class="fa fa-check"></i><b>4.5</b> Matrix Algebra</a>
<ul>
<li class="chapter" data-level="4.5.1" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#linear-combinations"><i class="fa fa-check"></i><b>4.5.1</b> Linear Combinations</a></li>
<li class="chapter" data-level="4.5.2" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#matrix-multiplication"><i class="fa fa-check"></i><b>4.5.2</b> Matrix multiplication</a></li>
<li class="chapter" data-level="4.5.3" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#matrix-inverses"><i class="fa fa-check"></i><b>4.5.3</b> Matrix Inverses</a></li>
<li class="chapter" data-level="4.5.4" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#transpose-and-hermitian-conjugate"><i class="fa fa-check"></i><b>4.5.4</b> Transpose and Hermitian Conjugate</a></li>
</ul></li>
<li class="chapter" data-level="4.6" data-path="matrices-and-linear-maps.html"><a href="matrices-and-linear-maps.html#orthogonal-and-unitary-matrices"><i class="fa fa-check"></i><b>4.6</b> Orthogonal and Unitary Matrices</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html"><i class="fa fa-check"></i><b>5</b> Determinants and inverses</a>
<ul>
<li class="chapter" data-level="5.1" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#introduction-1"><i class="fa fa-check"></i><b>5.1</b> Introduction</a></li>
<li class="chapter" data-level="5.2" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#epsilon-and-alternating-forms"><i class="fa fa-check"></i><b>5.2</b> <span class="math inline">\(\epsilon\)</span> and Alternating Forms</a>
<ul>
<li class="chapter" data-level="5.2.1" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#epsilon-and-permutation"><i class="fa fa-check"></i><b>5.2.1</b> <span class="math inline">\(\epsilon\)</span> and Permutation</a></li>
<li class="chapter" data-level="5.2.2" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#alternating-forms-and-linear-independence"><i class="fa fa-check"></i><b>5.2.2</b> Alternating Forms and Linear (In)dependence</a></li>
</ul></li>
<li class="chapter" data-level="5.3" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#determinants-in-mathbbrn-and-mathbbcn"><i class="fa fa-check"></i><b>5.3</b> Determinants in <span class="math inline">\(\mathbb{R}^n\)</span> and <span class="math inline">\(\mathbb{C}^n\)</span></a>
<ul>
<li class="chapter" data-level="5.3.1" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#definition"><i class="fa fa-check"></i><b>5.3.1</b> Definition</a></li>
<li class="chapter" data-level="5.3.2" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#evaluating-determinants-expanding-by-rows-or-columns"><i class="fa fa-check"></i><b>5.3.2</b> Evaluating determinants: expanding by rows or columns</a></li>
<li class="chapter" data-level="5.3.3" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#simplifying-determinants-row-and-column-operations"><i class="fa fa-check"></i><b>5.3.3</b> Simplifying determinants: Row and Column Operations:</a></li>
<li class="chapter" data-level="5.3.4" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#multiplicative-property"><i class="fa fa-check"></i><b>5.3.4</b> Multiplicative Property</a></li>
</ul></li>
<li class="chapter" data-level="5.4" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#minors-cofactors-and-inverses"><i class="fa fa-check"></i><b>5.4</b> Minors, Cofactors and Inverses</a>
<ul>
<li class="chapter" data-level="5.4.1" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#cofactors-and-determinants"><i class="fa fa-check"></i><b>5.4.1</b> Cofactors and Determinants</a></li>
<li class="chapter" data-level="5.4.2" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#adjugates-and-inverses"><i class="fa fa-check"></i><b>5.4.2</b> Adjugates and Inverses</a></li>
</ul></li>
<li class="chapter" data-level="5.5" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#system-of-linear-equations"><i class="fa fa-check"></i><b>5.5</b> System of Linear Equations</a>
<ul>
<li class="chapter" data-level="5.5.1" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#introduction-and-nature-of-solutions"><i class="fa fa-check"></i><b>5.5.1</b> Introduction and Nature of Solutions</a></li>
<li class="chapter" data-level="5.5.2" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#geometric-interpretation-in-mathbbr3"><i class="fa fa-check"></i><b>5.5.2</b> Geometric Interpretation in <span class="math inline">\(\mathbb{R}^3\)</span></a></li>
<li class="chapter" data-level="5.5.3" data-path="determinants-and-inverses.html"><a href="determinants-and-inverses.html#gaussian-elimination-and-echelon-form"><i class="fa fa-check"></i><b>5.5.3</b> Gaussian Elimination and Echelon Form</a></li>
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</ul></li>
<li class="chapter" data-level="6" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html"><i class="fa fa-check"></i><b>6</b> Eigenvalues and Eigenvectors</a>
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<li class="chapter" data-level="6.1" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#introduction-2"><i class="fa fa-check"></i><b>6.1</b> Introduction</a>
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<li class="chapter" data-level="6.1.1" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#definitions-3"><i class="fa fa-check"></i><b>6.1.1</b> Definitions</a></li>
<li class="chapter" data-level="6.1.2" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#examples-1"><i class="fa fa-check"></i><b>6.1.2</b> Examples</a></li>
<li class="chapter" data-level="6.1.3" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#deductions-involving-chi_at"><i class="fa fa-check"></i><b>6.1.3</b> Deductions involving <span class="math inline">\(\chi_A(t)\)</span></a></li>
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<li class="chapter" data-level="6.2" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#eigenspaces-and-multiplicities"><i class="fa fa-check"></i><b>6.2</b> Eigenspaces and Multiplicities</a>
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<li class="chapter" data-level="6.2.1" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#definitions-4"><i class="fa fa-check"></i><b>6.2.1</b> Definitions</a></li>
<li class="chapter" data-level="6.2.2" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#examples-2"><i class="fa fa-check"></i><b>6.2.2</b> Examples</a></li>
<li class="chapter" data-level="6.2.3" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#linear-independence-of-eigenvectors"><i class="fa fa-check"></i><b>6.2.3</b> Linear Independence of Eigenvectors</a></li>
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<li class="chapter" data-level="6.3" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#diagonalisability-and-similarity"><i class="fa fa-check"></i><b>6.3</b> Diagonalisability and Similarity</a>
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<li class="chapter" data-level="6.3.1" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#introduction-3"><i class="fa fa-check"></i><b>6.3.1</b> Introduction</a></li>
<li class="chapter" data-level="6.3.2" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#criteria-for-diagonalisability"><i class="fa fa-check"></i><b>6.3.2</b> Criteria for Diagonalisability</a></li>
<li class="chapter" data-level="6.3.3" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#similarity"><i class="fa fa-check"></i><b>6.3.3</b> Similarity</a></li>
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<li class="chapter" data-level="6.4" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#hermitian-and-symmetric-matrices"><i class="fa fa-check"></i><b>6.4</b> Hermitian and Symmetric Matrices</a>
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<li class="chapter" data-level="6.4.1" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#real-eigenvalues-and-orthogonal-eigenvectors"><i class="fa fa-check"></i><b>6.4.1</b> Real Eigenvalues and Orthogonal Eigenvectors</a></li>
<li class="chapter" data-level="6.4.2" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#unitary-and-orthogonal-diagonalisation"><i class="fa fa-check"></i><b>6.4.2</b> Unitary and Orthogonal Diagonalisation</a></li>
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<li class="chapter" data-level="6.5" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#quadratic-forms"><i class="fa fa-check"></i><b>6.5</b> Quadratic Forms</a>
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<li class="chapter" data-level="6.5.1" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#examples-in-mathbbr2-and-mathbbr3"><i class="fa fa-check"></i><b>6.5.1</b> Examples in <span class="math inline">\(\mathbb{R}^2\)</span> and <span class="math inline">\(\mathbb{R}^3\)</span></a></li>
</ul></li>
<li class="chapter" data-level="6.6" data-path="eigenvalues-and-eigenvectors.html"><a href="eigenvalues-and-eigenvectors.html#cayley-hamilton-theorem"><i class="fa fa-check"></i><b>6.6</b> Cayley-Hamilton Theorem</a></li>
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<li class="chapter" data-level="7" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html"><i class="fa fa-check"></i><b>7</b> Changing Bases, Canonical Forms and Symmetries</a>
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<li class="chapter" data-level="7.1" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#changing-bases-in-general"><i class="fa fa-check"></i><b>7.1</b> Changing Bases in General</a>
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<li class="chapter" data-level="7.1.1" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#definition-and-proposition"><i class="fa fa-check"></i><b>7.1.1</b> Definition and Proposition</a></li>
<li class="chapter" data-level="7.1.2" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#proof-of-proposition"><i class="fa fa-check"></i><b>7.1.2</b> Proof of proposition</a></li>
<li class="chapter" data-level="7.1.3" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#approach-using-vector-components"><i class="fa fa-check"></i><b>7.1.3</b> Approach using vector components</a></li>
<li class="chapter" data-level="7.1.4" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#comments-1"><i class="fa fa-check"></i><b>7.1.4</b> Comments</a></li>
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<li class="chapter" data-level="7.2" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#jordan-canonical-normal-form"><i class="fa fa-check"></i><b>7.2</b> Jordan Canonical/ Normal Form</a></li>
<li class="chapter" data-level="7.3" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#conics-and-quadrics"><i class="fa fa-check"></i><b>7.3</b> Conics and Quadrics</a>
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<li class="chapter" data-level="7.3.1" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#quadrics-in-general"><i class="fa fa-check"></i><b>7.3.1</b> Quadrics in General</a></li>
<li class="chapter" data-level="7.3.2" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#conics"><i class="fa fa-check"></i><b>7.3.2</b> Conics</a></li>
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<li class="chapter" data-level="7.4.1" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#orthogonal-transformations-and-rotations-in-mathbbrn"><i class="fa fa-check"></i><b>7.4.1</b> Orthogonal Transformations and Rotations in <span class="math inline">\(\mathbb{R}^n\)</span></a></li>
<li class="chapter" data-level="7.4.2" data-path="changing-bases-canonical-forms-and-symmetries.html"><a href="changing-bases-canonical-forms-and-symmetries.html#d-minkowski-space-and-lorentz-transformations"><i class="fa fa-check"></i><b>7.4.2</b> 2d Minkowski Space and Lorentz Transformations</a></li>
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<h1 class="title">Vectors and Matrices 2021 - 2022</h1>
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<p class="date"><em>07/10/2021</em></p>
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<h1>Lecture 1</h1>
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<h2><span class="header-section-number">0.1</span> Contents</h2>
<ol style="list-style-type: decimal">
<li>Complex Numbers</li>
<li>Vectors in 3d</li>
<li>Vectors in General, <span class="math inline">\(\mathbb{R}^n \times \mathbb{C}^n\)</span></li>
<li>Matrices and Linear Maps</li>
<li>Determinants and Inverses</li>
<li>Eigenvalues and Eigenvectors</li>
<li>Changing Bases, Canonical Forms and Symmetries</li>
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