Mean as a Projection

This tutorial explains how mean can be viewed as an orthogonal projection onto a subspace defined by the span of an all 1’s vector (i.e., basis vector). Suppose that \( \vec{y} \in \mathbb{R}^n \) and \( L \subset \mathbb{R}^n\) is the span defined by the space of vector \( \vec{x} \), namely, \( \vec{x}=\left[\begin{array}{ccc}1\\1\\ …\\ … Read more

Orthogonal Projection

This tutorial explains what an orthogonal projection is in linear algebra. Further, it provides proof that the difference between a vector and a subspace is orthogonal to that subspace. Let’s define two vectors, \(\vec{X} \) and \(\vec{Y} \), and we want to find the shortest distance between \(\vec{Y} \) and the subspace defined by the … Read more

Difference between Space and Subspace

This tutorial explains the difference between space and subspace. What is space? The most simple example of space is the two-dimensional space, \( \mathbb{R}^2 \)​​. You can visualize it as the xy-coordinate plane. \( \mathbb{R}^2 \)​​ is​​ a vector space that contains all possible two-dimensional vectors \( \vec{v} = (x, y) \). Similarly, \( \mathbb{R}^3 … Read more

Linear Subspace

A linear subspace or vector subspace is a vector space that is a subset of some larger vector space. To be considered a linear subspace, a vector set needs to meet the following 3 requirements. For instance, if we say that \( V \) is a subset of \( \mathbb{R}^n \) and in order to … Read more

Vector and Arrow in Space

This tutorial provides examples to explain vectors and arrows in space (vector visualization). It includes 2 and 3-dimension vectors as well as vector addition and subtraction. Example 1: 2 dimension vectors \( \vec{V_1} =(3, 2 ) = \left[\begin{array}{ccc}3\\2 \end{array}\right]\) Note that, \( (3, 2 ) \) and \( \left[\begin{array} {ccc} 3\\ 2 \end{array} \right]\) are … Read more

Orthonormal Vectors: Definitions and Examples

Two Orthogonal Vectors Definition: Two vectors are orthogonal if they are perpendicular to each other. That is, the dot product of the two vectors is zero. The following is an example of two orthonormal vectors. \( \vec{V_1} =\left[\begin{array}{ccc}1\\0\\-1\end{array}\right]\), \( \vec{V_2} =\left[\begin{array}{ccc}1\\3\\1\end{array}\right] \) That is, \( (1 \times 1) + (0 \times 3) +(-1 \times 1) … Read more