Manitoba linear algebra

Subsection 4.1.1 The definitions of Euclidean \(n\)-space

Euclidean \(n\) space, also called \(\R^n\text{,}\) is formally written as

This means that it consists of elements called \(n\)-tuples, which are written as \((x_1,x_2,\ldots,x_n)\) where each \(x_i\) is a real number. Each such element is called a vector.

The most familiar examples, of course, are \(\R^2\text{,}\) the plane, and \(\R^3\text{,}\) ordinary 3-dimensional space. For \(\R^2\) we have an \(x\)-axis and a \(y\)-axis, and the points in the plane as \(2\)-tuples are defined by dropping perpendiculars to each axis.

Euclidean \(3\)-space is viewed analogously.

There are, of course, many geometric concepts studied in \(\R^2\) and \(\R^3\text{.}\) One of our goals is to see how these concepts can be extended to \(\R^n\text{.}\)

While Euclidean \(n\)-space consists of \(n\)-tuples, they are sometimes viewed from different mathematical perspectives.

• Points in \(n\)-space: the vectors are just the \(n\)-tuples \((x_1,x_2,\ldots,x_n)\text{.}\)

• Directed vectors: the vectors may be thought of as arrows from an initial point \(P\) to a terminal point \(Q\text{.}\)

• Column vectors where the vector is an \(n\times 1\) matrix:

  \begin{equation*} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix} \end{equation*}

• Row vectors where the vector is a \(1\times n\) matrix:

  \begin{equation*} \begin{bmatrix} x_1, x_2, \ldots, x_n \end{bmatrix} \end{equation*}

For our initial discussion, we will concentrate on \(n\)-tuples and column vectors.

Subsection 4.1.2 Equality, addition and scaler multiplication of vectors

The equality, addition and scalar multiplication of \(n\)-tuples is very much like that of matrices:

• Equality: \((x_1,x_2,\ldots,x_n)=(y_1,y_2,\ldots,y_n)\) means \(x_i=y_i\) for \(i=1,2,\ldots,n\text{.}\)

• Addition: \((x_1,x_2,\ldots,x_n)+(y_1,y_2,\ldots,y_n) =(x_1+y_1,x_2+y_2,\ldots,x_n+y_n)\text{.}\)

• Scalar multiplication: For any scalar \(r\text{,}\) \(r(x_1,x_2,\ldots,x_n) =(rx_1,rx_2,\ldots,rx_n)\text{.}\)

In fact, if we look at the \(n\)-tuples as column vectors, then equality, addition and scalar multiplication are the same as matrix equality, addition and scalar multiplication.