Linear Algebra Fundamentals

Linear algebra is the branch of mathematics concerning linear equations, linear maps, and their representations in vector spaces and through matrices.

Vectors

A <strong>vector</strong> is a quantity with both magnitude and direction. In n-dimensional space, a vector can be represented as an ordered list of n numbers.

v = [v₁, v₂, v₃, ..., vₙ]

Vector Operations

<li>&amp;lt;strong&amp;gt;Addition&amp;lt;/strong&amp;gt;: Vectors are added component-wise</li> <li>&amp;lt;strong&amp;gt;Scalar multiplication&amp;lt;/strong&amp;gt;: Each component is multiplied by the scalar</li> <li>&amp;lt;strong&amp;gt;Dot product&amp;lt;/strong&amp;gt;: Sum of products of corresponding components</li>

Matrices

A <strong>matrix</strong> is a rectangular array of numbers arranged in rows and columns.

<thead> &lt;tr&gt; &amp;lt;th&amp;gt;Operation&amp;lt;/th&amp;gt; &amp;lt;th&amp;gt;Description&amp;lt;/th&amp;gt; &lt;/tr&gt; </thead> <tbody> &lt;tr&gt; &amp;lt;td&amp;gt;Addition&amp;lt;/td&amp;gt; &amp;lt;td&amp;gt;Add corresponding elements&amp;lt;/td&amp;gt; &lt;/tr&gt; &lt;tr&gt; &amp;lt;td&amp;gt;Multiplication&amp;lt;/td&amp;gt; &amp;lt;td&amp;gt;Row × Column dot products&amp;lt;/td&amp;gt; &lt;/tr&gt; &lt;tr&gt; &amp;lt;td&amp;gt;Transpose&amp;lt;/td&amp;gt; &amp;lt;td&amp;gt;Flip rows and columns&amp;lt;/td&amp;gt; &lt;/tr&gt; </tbody>

Key Concepts

<p>"The essence of mathematics lies in its freedom." — Georg Cantor</p>

Eigenvalues and Eigenvectors

For a square matrix A, if:

Av = λv

Then λ is an <strong>eigenvalue</strong> and v is the corresponding <strong>eigenvector</strong>.

Applications

<li>Computer graphics transformations</li> <li>Machine learning (PCA, SVD)</li> <li>Quantum mechanics</li> <li>Network analysis</li>