\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Linear algebra can generally be defined as the study of vectors. The algebra which most people are familiar with defines the rules for solving equations involving numbers. Linear algebra defines the rules for manipulating equations where the variables are vectors.

Vectors can be thought of as an ordered list of numbers. They are written with an arrow over their name and their scalar components are usually written in vertical brackets. The components of a vector can also be referenced according to their index through the use of subscripts.

Example of vectors:

\( \overrightarrow{a} = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}\quad \) \( \overrightarrow{b} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}\quad \) \( \overrightarrow{d} = \begin{bmatrix} d_1 \\ d_2 \\ \vdots \\ d_n \end{bmatrix} \)

If the vector \( \overrightarrow{r} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \) then its components can be referenced according to their position in the vector. We use the name of the vector with a subscript to indicate the components position/index in the vector, \( r_1 = 3 \) and \( r_2 = 4 \)

Formally defined a vector space is a set which satisfies the axioms defining a vector space. Informally a vector space can be thought of as a set which contains all vectors of a specific dimension.

The vector space R1 contains all “1-D” vectors. These vectors consist of 1 scalar component, a real number, and are visualized as living in the real number line. The tail of these vectors start at the origin (0) and their head lies at the point(x) specified by its components. Because all vectors belonging to the vector space R1 lie on the real number line, the vector space itself can be thought of as the real number line.

To the left is a visualization of the vector \( \overrightarrow{a} \) which belongs to the vector space \( \mathbb{R^1} \). **Edit \( \overrightarrow{a} \) by sliding the red slider below.**

The vector space R2 contains all “2-D” vectors. These vectors consist of 2 scalar components, two real numbers, and are visualized as living in the real plane. The tail of these vectors start at the origin (0,0) and their head lies at the point (x,y) specified by its components. Because all vectors belonging to the vector space R2 lie on the real plane, the vector space itself can be thought of as the real plane.

To the left is a visualization of the vector \( \overrightarrow{b} \) which belongs to the vector space \( \mathbb{R^2} \). **Edit \( \overrightarrow{b} \) by sliding the red and green sliders below.**

The vector space R3 contains all “3-D” vectors. These vectors consist of 3 scalar components, three real numbers, and are visualized as living in 3D space. The tail of these vectors start at the origin (0,0,0) and their head lies at the point (x,y,z) specified by its components. Because all vectors belonging to the vector space R3 live in 3D space, the vector space itself can be thought of as 3D space.

To the left is a visualization of the vector \( \overrightarrow{c} \) which belongs to the vector space \( \mathbb{R^3} \). **Edit \( \overrightarrow{c} \) by sliding the red, green and blue sliders below.**

Although R1, R2,and R3 will be the only vector spaces we will deal with in this series they are by no means the only Vector Spaces possible.

The reason we will focus on the vectors spaces R1, R2, and R3 is that many topics in linear algebra can be more readily understood when we are able to apply our spatial reasoning to develop and motivate central concepts in the subject.

By restricting ourselves to vector spaces which are easily visualized I hope to help students who may be struggling with the subject by introducing the topics in a visual manner through the use of interactive visualizations.

The rules and concepts of linear algebra are used to analyze and model systems with Vector Spaces which do not necessarily lend themselves to visualizations. Some examples of such vector spaces include

- a. Complex Valued Vector Space
- b. The Vector Space of \( m \times n \) Matrices
- c. The Vector Space of Real Valued Functions

Linear Algebra defines the rules for how to deal with general/abstract vector spaces, and readers of this series should limit themselves to think of Linear Algebra as only being the study of vectors we can visualize in our familiar 3 dimensions.