Infinite number of variables and infinite number of equations

Question

Consider I have a system of linear equations $Ax=b$, where $A$ is a (countable) infinite by infinite matrix, $x$ and $b$ are infinite by 1 vector. If $A$ has infinite many zero rows, can we say the number of variables is larger than the number of equations this case? What is the degree of freedom in this case?

Answer 1

If $A$ has infinite many zero rows, can we say the number of variables is larger than the number of equations this case?

Most certainly not. Take for example the infinite matrix $A$ defined by $$A_{ij}=\begin{cases}1&\text{if } i=2\cdot j\\ 0&\text{otherwise}\end{cases}$$

which is the matrix

$$\begin{bmatrix}0&0&0&\dots\\ 1 & 0&0&\dots\\0&0&0&\dots\\0&1&0&\dots\\\vdots&\vdots&\vdots&\ddots\end{bmatrix}$$

In this case, not only is the "number" of nonzero matrix countably infinite (i.e., the same as the "number" of values of $x$), also, if you take $$b=\begin{bmatrix}0\\x_1\\0\\x_2\\\vdots\end{bmatrix}$$

then the system has a unique solution $x=b$.

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Degrees of freedom are not a well defined term for infinite matrices.