Scalars and Vectors in n-Dimensional Space

The Anatomy of a Vector

Let's begin by asking a simple question: what is a vector? The answer depends on your perspective, and linear algebra beautifully unifies these different viewpoints.

A great way to start is by watching the first video from the "Essence of linear algebra" series by 3Blue1Brown. It introduces the main ways people think about vectors.

As the video explains, a physicist might see a vector as an arrow in space, defined by its length and direction. A computer scientist might see it as an ordered list of numbers. A mathematician seeks to generalize both. In this course, we will primarily use the computer science and physics perspectives, as they are most concrete.

The key to connecting these ideas is the coordinate system. For our purposes in linear algebra, we will almost always consider vectors as arrows whose tails are fixed at the origin (the point (0,0) in 2D or (0,0,0) in 3D). This is a crucial convention that simplifies many operations.

The video continues by formalizing this convention.

With the vector's tail at the origin, its coordinates precisely describe the location of its tip. This gives us a powerful one-to-one correspondence: every arrow from the origin corresponds to a unique list of numbers, and every list of numbers corresponds to a unique arrow from the origin.

For example, in a 2D plane, the vector $\vec{v}=\left[\begin{array}{c}3\\ 2\end{array}\right]$ is an arrow that starts at the origin (0,0) and ends at the point (3,2). The numbers in the vector are its components. The first component tells us how far to move along the horizontal x-axis, and the second tells us how far to move along the vertical y-axis.

A vector $\vec{v}$ in the 2D plane, represented by the coordinates $\left[\begin{array}{c}x\\ y\end{array}\right]$. Its tail is at the origin (0,0) and its tip is at the point (x,y).

We denote vectors with either a small arrow on top, like $\vec{v}$, or as a bold lowercase letter, like v. In this course, we will follow the standard convention of writing vectors as column vectors (or column matrices), like the $\left[\begin{array}{c}3\\ 2\end{array}\right]$ example above.

From 2D and 3D to n-Dimensional Space

Visualizing vectors in two or three dimensions is intuitive. But what about vectors with four, five, or even a hundred components? We can't draw a picture of a 100-dimensional arrow. This is where the "list of numbers" perspective becomes essential.

The collection of all possible vectors with $n$ real-numbered components is called n-dimensional space, which we denote as ${\mathbb{R}}^n$ (pronounced "R-n").

• ${\mathbb{R}}^1$ is the set of all vectors with one component, like $[x]$. This is simply the set of all real numbers, which we can visualize as the number line.
• ${\mathbb{R}}^2$ is the set of all vectors with two components, like $\left[\begin{array}{c}x\\ y\end{array}\right]$. We visualize this as the 2D Cartesian plane.
• ${\mathbb{R}}^3$ is the set of all vectors with three components, like $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$. We visualize this as the 3D space we live in.

For dimensions higher than three, we lose the direct geometric picture, but the algebraic idea remains perfectly valid. A vector in ${\mathbb{R}}^n$ is simply an ordered list, or n-tuple, of $n$ real numbers. For example, the vector below is an element of ${\mathbb{R}}^5$:

Each house in the 3Blue1Brown video's real estate example, described by its square footage and price, could be considered a vector in ${\mathbb{R}}^2$. If we also included the number of bedrooms and the age of the house, each house would be represented by a vector in ${\mathbb{R}}^4$.

The following video provides a clear, formal definition of ${\mathbb{R}}^n$.

Defining Scalars

Now that we have a solid grasp of vectors, let's turn to the other key term for this lesson: scalars.

In the context of linear algebra, a scalar is simply a real number.

You might wonder why we need a special name for something as familiar as a number. The term "scalar" is used to emphasize the number's role in an operation called scalar multiplication. When we multiply a vector by a scalar, we scale the vector.

Let's return to the 3Blue1Brown video to see what this means geometrically.

As the video demonstrates, multiplying a vector by a scalar $c$ has a clear geometric effect:

• If $c>1$, the vector is stretched.
• If $0<c<1$, the vector is squished.
• If $c<0$, the vector is flipped to point in the opposite direction and then scaled by the magnitude of $c$.

Numerically, the operation is straightforward: you multiply every component of the vector by the scalar. For a vector $\vec{v}=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]$ in ${\mathbb{R}}^n$ and a scalar $c$, the scalar product is:

For instance, if $\vec{v}=\left[\begin{array}{c}1\\ -2\end{array}\right]$ and the scalar is $c=3$, then:

Geometrically, the vector $\left[\begin{array}{c}3\\ -6\end{array}\right]$ points in the same direction as $\left[\begin{array}{c}1\\ -2\end{array}\right]$ but is three times as long.

Conclusion

In this introductory lesson, we have established the fundamental concepts of vectors and scalars, which are the atoms and molecules of linear algebra.

Here are the key takeaways:

• Vectors can be understood geometrically as arrows with a specific length and direction, or algebraically as ordered lists of numbers (components). In linear algebra, we typically anchor these arrows at the origin of a coordinate system.
• The space containing all possible vectors with $n$ real components is called n-dimensional space, denoted ${\mathbb{R}}^n$. A vector with $n$ components is an element of ${\mathbb{R}}^n$.
• A scalar is simply a real number. Its name comes from its role in scalar multiplication, where it scales a vector by stretching, shrinking, and/or reversing its direction. This is done by multiplying each component of the vector by the scalar.

You now have the basic vocabulary and conceptual framework needed to proceed. In our next lesson, we will build directly on this foundation by exploring the two fundamental vector operations: addition and scalar multiplication. You've already had a preview of the latter, and soon you'll see how these two operations together unlock the power to combine and manipulate vectors in powerful ways.