29 Vectors

Vectors in two-dimensions are made up of an initial point and a terminal point. In the xy-coordinate plane, the initial point is at (0,0), and the terminal point is at some other coordinate (x,y) [16]. If we have a vector

    \[\textbf{v}=(x_1,x_2),\]

then (x_1,x_2) is the terminal point, and x_1 and x_2 are the components [16]. One more important fact is that a vector is denoted with a bold lowercase letter [16]. Vectors do not have to be two-dimensional. In fact, they can have as many dimensions as we want. A vector in n-dimensions is denoted as \textbf{v}=(x_1, x_2, ..., x_n) [16].

Two basic operations can be performed on vectors, vector addition and scalar multiplication. Vector addition is done component-wise. That is, for two vectors \textbf{u}=(x_1,x_2,...,x_n) and \textbf{v}=(y_1, y_2, ..., y_n) in n dimensions,

    \[\textbf{u}+\textbf{v}=(x_1+y_1,x_2+y_2,...,x_n+y_n).[16]\]

For a vector \textbf{u}=(x_1,x_2,...,x_n) and scalar c, scalar multiplication is defined as

    \[c\textbf{u}=(cx_1,cx_2,...,cx_n). [16]\]

As you can see, these operations are performed in much the same way as matrix addition and scalar multiplication of matrices. This makes sense as matrices are a subset of vectors.

We have a great amount of information to discuss relating to vectors so let’s begin our next section on vector spaces.

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