How do spin operators work? [closed]

I am currently studying statistics and 2D Ising models and noticed in my lecturer's notes the operators, acting in the spin space

The text says that this is identity $$2^N\times 2^N$$ matrix. I don't understand how this operator acts on the vector of spins. Usually there is only tensor convolution by index, but here are fixed spins in a row, for example:

$$\{\uparrow,\uparrow,\downarrow\ldots,\uparrow\}$$

How should I change their value if they are fixed?

Someone, please explain the action, for example, for the simple operator $$\delta_{\sigma_1'\sigma_1}$$ and for spin row $$\{\uparrow\}$$. Should it work like this? $$\delta_{\sigma_1'\sigma_1}\{\sigma_1\}=\delta_{\sigma_1'\uparrow}\{\uparrow\}+\delta_{\sigma_1'\downarrow}\{\downarrow\}$$

What I've even got? And also how can this matrix be $$2^N\times 2^N$$ if the row is made of $$N$$ spins? I am probably too sleepy and just forgot how tensors work...

Ok, probably I guessed how it should work. At first we have only one vector in our space where operator act. This is not the vector of spins $${\sigma_1,\sigma_2,\ldots,\sigma_n}$$. This is the vector of all possible states (every element of this vector is spin vector) with any order that we will fix initially (equivalence class), for example for 3 spins: $$\vec{s}=\left(\begin{array}{c} \{\uparrow,\uparrow,\uparrow\} \\ \{\uparrow,\uparrow,\downarrow\} \\ \{\uparrow,\downarrow,\uparrow\} \\ \{\uparrow,\downarrow,\downarrow\} \\ \{\downarrow,\uparrow,\uparrow\} \\ \{\downarrow,\uparrow,\downarrow\} \\ \{\downarrow,\downarrow,\uparrow\} \\ \{\downarrow,\downarrow,\downarrow\} \\ \end{array}\right)=\left(\begin{array}{c} \vec{\sigma}[1] \\ \vec{\sigma}[2] \\ \vec{\sigma}[3] \\ \vec{\sigma}[4] \\ \vec{\sigma}[5] \\ \vec{\sigma}[6] \\ \vec{\sigma}[7] \\ \vec{\sigma}[8] \\ \end{array}\right)$$ And in this case matrix $$2^3\times 2^3$$ will correctly work: $$\vec{s'}=E_{8\times 8}\left(\begin{array}{c} \{\uparrow,\uparrow,\uparrow\} \\ \{\uparrow,\uparrow,\downarrow\} \\ \{\uparrow,\downarrow,\uparrow\} \\ \{\uparrow,\downarrow,\downarrow\} \\ \{\downarrow,\uparrow,\uparrow\} \\ \{\downarrow,\uparrow,\downarrow\} \\ \{\downarrow,\downarrow,\uparrow\} \\ \{\downarrow,\downarrow,\downarrow\} \\ \end{array}\right)=\left(\begin{array}{c} \{\uparrow,\uparrow,\uparrow\} \\ \{\uparrow,\uparrow,\downarrow\} \\ \{\uparrow,\downarrow,\uparrow\} \\ \{\uparrow,\downarrow,\downarrow\} \\ \{\downarrow,\uparrow,\uparrow\} \\ \{\downarrow,\uparrow,\downarrow\} \\ \{\downarrow,\downarrow,\uparrow\} \\ \{\downarrow,\downarrow,\downarrow\} \\ \end{array}\right)=\vec{s}$$ So we numbered the sequence of states (spinvectors) and marked elements as $$\vec{\sigma}[i]$$. It's not the component of $$\vec{\sigma}$$, it's whole spinvector at the i-th place of sequence. We want to write: $$\vec{\sigma}'[i]=\delta_{ij}\vec{\sigma}[j]$$ And then to move from terms of $$\vec{\sigma}[i]$$ to normal spinvectors with their own indexes. So we have to number states not by artificial order but by their own spins. In terms of components, for example, for 2 spins (sigmavector has two compoments) we can write: $$\sigma'_{ik}=\delta_{ij}\delta_{kl}\sigma_{jl}$$ Let's check it: $$\sigma'_{\uparrow\uparrow}=\delta_{\uparrow j}\delta_{\uparrow l}\sigma_{jl}=\sigma_{\uparrow\uparrow}$$ And the same logic with another indexes, it really works like identity operator. It can look absurd for some operators, because we can get smth like $$\sigma'_{\uparrow\uparrow}=\uparrow\downarrow$$ but as we saw earlier the index $$_{\uparrow\uparrow}$$ is only necessary to mark the order of spinvectors in $$\vec{s}$$. It will only mean that the state where spins were $$\uparrow\uparrow$$ has now spins $$\uparrow\downarrow$$.
May be I couldn't understand the stupid stuff but teachers notes looked like they supposed acting on spinvectors $$1\times N$$ and it confused me.