# Standard Basis of the Finite Field of Prime Numbers

A little info regarding this field:

Addition and multiplication in $Z^n_p$ behave as usual but with the remainder taken upon division by $p$.

Ex: $Z_3$ will only consist of the three integers {$0,1,2$}.

• $2+1=0$
• $2+2=1$
• $2*2=1$
• $Z_3^3 =\begin{Bmatrix} (0,0,0),(1,0,0),...(2,2,2)\end{Bmatrix}$
• $dim(Z_3^3)=3^3=27$

What I'm unure of is the standard basis associated with this field.

Contextually, I'm trying to find the image of a transformation: $T:Z_3^3\rightarrow Z^2_3$

Defined by $T(x)=Ax$ where $A=\begin{pmatrix}1 & 0 & 2\\ 1 & 2 & 1\end{pmatrix}$

From that I know that I can find the image by finding the largest linearly independent subset of T applied to each element of the standard basis and that will be the basis for $im(T)$.

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There is a field with 27 elements, but you haven't described it here. Do you mean the ring $\mathbb Z_3^3$? Or do you really mean the finite field $GF(27)$? If you are not sure, then tell us how (or whether) you propose to multiply two elements of your structure. – TonyK Feb 2 '13 at 18:27

If $F$ is any field, the standard basis of the $F$-vector space $F^n$ is always the set $$\{e_1=(1,0,0,\ldots,0),e_2=(0,1,0,\ldots,0),\ldots,e_n=(0,0,\ldots,0,1)\}$$
As for your underlying question, "find the largest linearly independent subset of T applied to each element of the standard basis" just means "consider the space spanned by the columns of $A$". Because the codomain of $T$ is two-dimensional, and $A$ has at least two columns that are linearly independent (which because there are only two just means that neither is a multiple of the other other), the image of $T$ is the entire $F^2$.