Did my Linear algebra textbook make a mistake here or am I misunderstanding something?

Question

So, it is my understanding that bold text indicates a vector. Why are a$_1$ a$_2$.... a$_n$ listed in bold text? Are a$_1$ a$_2$.... a$_n$ not referring to columns which only "consist" of one variable and therefore are not vectors? I thought that each row, not column, of the matrix A was a vector.

plese help me with what i am not understanding, thanks.

Answer 1

Probably, your confusion arises because you think of the vector $(2,3)$ as going "$2$ units in the $x$ direction and $3$ units in the $y$ direction", so something like $$\begin{bmatrix}\mathbf{a}_1&\mathbf{a}_2&\cdots&\mathbf{a}_n\end{bmatrix}$$ confuses you since you cannot go "$\mathbf{a}_1$ units in the direction of the first variable". The problem here is that thinking about vectors as "going a certain distance in a certain direction" only works when the vectors we consider have real components. In general however, the components of a vector can be anything from a complex number to another vector of real entries (they can come from any vector space). In this case, we can think of a matrix as a vector of vectors: for instance with $$\mathbf{a}_1=\begin{bmatrix}1\\0\end{bmatrix},\mathbf{a}_2=\begin{bmatrix}0\\1\end{bmatrix},\mathbf{a}_3=\begin{bmatrix}1\\1\end{bmatrix},$$ we have $$\begin{bmatrix}\mathbf a_1&\mathbf a_2&\mathbf a_3\end{bmatrix}=\begin{bmatrix}1&0&1\\0&1&1\end{bmatrix}$$ by simply writing the column vectors down together, concatenating them together in sequence.

Answer 2

This example should help you understand.

The matrix $\begin{bmatrix} 1&&2\\3&&4\end{bmatrix}$ has columns $\mathbf c_1=\begin{bmatrix}1\\3\end{bmatrix}$ and $\mathbf c_2=\begin{bmatrix}2\\4\end{bmatrix}.$

It could be written $\begin{bmatrix} \mathbf c_1&&\mathbf c_2\end{bmatrix}$.

Indeed, each row of a matrix can be thought of as a vector, and each column can be as well.

Answer 3

"There are two commonly used ways to denote the $n$-tuples of $\mathbb{R}^{n}$. As rows [...] or columns".

If your vectors are rows, then each row of your matrix $A$ is a vector. If your vectors are columns, then each column of your matrix $A$ is a vector.

In the matrix you give: $$A= \begin{bmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \cdots & \mathbf{a}_{n} \end{bmatrix}$$ each $\mathbf{a}_{i}$ represents a column vector, and represents a column of the matrix $A$.

You could just as easily use $\mathbf{r}_{1}$, $\ldots$, $\mathbf{r}_{m}$ to represent row vectors, and write $$A = \begin{bmatrix} \mathbf{r}_{1}\\ \mathbf{r}_{2} \\ \vdots \\\mathbf{r}_{m} \end{bmatrix}$$ where each $\mathbf{r}_{j}$ represents a row of the matrix $A$.