A Simple 3×2 Matrix

Consider the $3 \times 2$ matrix

$A = \begin{pmatrix} 3 & 4.5 \\ 2 & 1.2 \\-0.1 & 8.2 \end{pmatrix}.$

The matrix can be interpreted as the collection of two column vectors: $A=(a_{1}, a_{2})$ , where $a_{j}$ ‘s contain the columns of $A$ :

$a_{1}=\begin{pmatrix} 3\\ 2\\-0.1\end{pmatrix}, a_{2}=\begin{pmatrix} 4.5\\ 1.2\\8.2\end{pmatrix}.$

Geometrically, $A$ represents $2$ points in a $3$ -dimensional space.

Alternatively, we can interpret $A$ as a collection of 3-row vectors in $\mathbb{R}^2$ .

$A = \begin{pmatrix} b_{1}^{T}\\b_{2}^{T}\\b_{3}^{T} \end{pmatrix}.$

where $b_{i}$ ‘s, $i=1, 2, 3$ contain the rows of $A$ :

$b_{1}=\begin{pmatrix} 3\\ 4.5\end{pmatrix}, b_{2}=\begin{pmatrix} 2\\ 1.2\end{pmatrix}, b_{3}=\begin{pmatrix} -0.1\\ 8.2\end{pmatrix}.$

Geometrically, $A$ represents 3 points in a 2-dimensional space.

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