There are two major objects in Linear Algebra: vectors and matrices. Here, we will review some important basics with vectors. Most of this lesson is review from Calculus II.

Vectors are lists of numbers. Actually, there is a more technical definition we will get to much later in this course. But for now, vectors are lists of numbers. There are numerous notations used for vectors, including the following:

$$\large \vec{x}=\left<3,2,-1,5\right>$$

$$\large \vec{x} = 4\hat{i}-3\hat{j}+\hat{k}$$

$$ \large \vec{x}= \begin{bmatrix} 2\\ 0\\ 5\\ -12\\ 1\\ \end{bmatrix} $$

In this class, we may use any of the notations from above.

Defnition: Vectors belong to a set called a vector space. The vector space for these vectors is called $\large \mathbb{R}^n$. The $\large n$ here is the dimension of the vector space, and it's just the length of the vector.

The first vector listed above has four elements, so it belongs to $\large \mathbb{R}^4$. We write $\large \vec{x}\in R^4$, and we say $\large \vec{x}$ is in $\large \mathbb{R}^4$.

Defnition: A scalar is just a real or complex number.

Part of linear algebra is figuring out the rules of arithmetic for vectors and matrices. Can we add these things? Multiply? Divide? When do the traditional rules of operations applky? Do we have laws of commutivity? Associativity? Distributivity?

Addition and subtration of vectors is just done component by component:

$$\large \begin{bmatrix} 3\\ -4\\ 8\\ \end{bmatrix} + \begin{bmatrix} 1\\ 0\\ -1\\ \end{bmatrix} = \begin{bmatrix} 4\\ -4\\ 7\\ \end{bmatrix} $$

It is quite simple to multiply a scalar and a vector:

$$\large 2\begin{bmatrix} 2\\ -3\\ 7\\ \end{bmatrix} = \begin{bmatrix} 4\\ -6\\ 14\\ \end{bmatrix} $$

Definition: the dot product (also known as the inner product) of two vectors is the sum of the products of the corresponding components of the vectors.

That is,

$$\displaystyle \large \vec{x}\cdot\vec{y} = x_1y_1+x_2y_2+\dots+x_ny_n=\sum_{k=1}^n x_ky_k$$

The following definition will be used quite a bit in this class:

Definition: A sum of scaled versions of a collection of vectors is called a linear combination of those vectors.

The following are all different linear combinations of the vectors $\large \vec{x}_1$, $\large \vec{x}_2$, and $\large \vec{x}_3$:

$$\large 3\vec{x}_1-2\vec{x}_2+\vec{x}_3$$

$$\large 10\vec{x}_2-7\vec{x}_3$$

$$\large -\vec{x}_1+\vec{x}_2-2\vec{x}_3$$

In $\large \mathbb{R}^2$ and $\large \mathbb{R}^3$, we can picture vectors as arrows. It's nice to build intuition in these small dimensions before we move into larger vector spaces where much of linear algebra actually happens.

Here we see a graphical representation of the addition of the vectors $\large \left&lt;3,4\right&gt;$ and $\large \left&lt;-1,6\right&gt;$:

Thinking of vectors in this way leads to the idea of the length of a vector:

Definition: The norm, or length, of $\large \vec{x}$ is denoted by $\large \lVert\vec{x}\rVert$ and is calculated by

$$\large\displaystyle \lVert\vec{x}\rVert=\sqrt{x_1^2+x_2^2+\dots+x_n^2}$$

Relatedly, we can measure the angle between vectors:

$$\large \cos(\theta)=\frac{\vec{x}\cdot\vec{y}}{\lVert\vec{x}\rVert \lVert\vec{y}\rVert}$$

This leads to a very important fact for vectors - the dot product is our key to determine if two vectors are orthogonal:

Theorem: Two vectors are orthogonal (or perpendicular or normal, all three words mean the same thing) if their dot product equals zero.

After this this section, I can:

• Recall the definitions: scalar, linear combination, norm, and $\large \mathbb{R}^n$
• Compute linear combinations of vectors.
• Determine if vectors are orthogonal.
• Calculate the length of a vector, the distance between vectors, or the angle between vectors.