MIT 18.06SC Lecture 5.3: Vector Spaces

Context

My lecture notes

Vector spaces and subspaces are fundamental structures in linear algebra. This post covers the vector spaces portion of Lecture 5.

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Definition

The space \(\mathbb{R}^n\) has exactly n dimensions.

Example: \[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] is a matrix in \(\mathbb{R}^{2 \times 2}\) (2×2 real matrices).

Subspaces

A subspace is a subset of \(\mathbb{R}^n\) that is closed under addition and scalar multiplication.

Examples of Subspaces

In \(\mathbb{R}^2\): - A line through the origin creates a subspace

In \(\mathbb{R}^3\): - A line through the origin creates a line subspace - A plane through the origin creates a plane subspace

In \(\mathbb{R}^4\): - Think: How can we create subspaces in \(\mathbb{R}^4\)?

Properties of Subspaces

A subspace must satisfy two properties:

1. Closed Under Addition

If vectors \(\vec{v}\) and \(\vec{w}\) are in the subspace, then \(\vec{v} + \vec{w}\) is also in the subspace.

Example: For \(\mathbb{R}^2\) line \(y=2x\): - \([1,2]\) and \([2,4]\) are in the line - \([1,2] + [2,4] = [3,6]\) is still in the line ✓

2. Closed Under Scalar Multiplication

If vector \(\vec{v}\) is in the subspace and \(c\) is any scalar, then \(c\vec{v}\) is also in the subspace.

Example: For \(\mathbb{R}^2\) line \(y=2x\): - \([1,2]\) is in the line - \(1.5 \times [1,2] = [1.5, 3]\) is still in the line ✓

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Source: MIT 18.06SC Linear Algebra, Lecture 5