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Vector and Matrices

Vectors and Matrices

Vectors: The Building Blocks

At its core, a vector is just a list of data . Whether it’s a row of GDP numbers or a column of batting averages, vectors structure our data.

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Visualizing Vectors

In R2R^2 (2D space), a vector is an arrow starting from the origin (0,0)(0,0).

  • Magnitude: The length of the arrow.
  • Direction: Where the arrow points.

Understanding Matrices

A matrix is a rectangular array of numbers. It’s essentially a collection of vectors.

Types of Matrices

  1. Square Matrix: Same number of rows and columns (n×nn \times n).
  2. Diagonal Matrix: Non-zero entries only on the diagonal.
  3. Scalar Matrix: A diagonal matrix where all diagonal elements are equal.
  4. Identity Matrix (II): A scalar matrix with 11s on the diagonal.

Matrix Operations

  • Addition: Add corresponding elements.
  • Scalar Multiplication: Multiply every element by the scalar.
  • Matrix Multiplication: The row of the first matrix dots with the column of the second.

Determinants

The determinant is a special number calculated from a square matrix.

  • Invariance: Adding a multiple of one row to another does not change the determinant.
  • Minors & Cofactors: Tools used to calculate determinants of higher-order matrices.