At the very beginning of the "Algebraic Supplement" of "Number Theory" (Borevich-Shafarevich) I read:
Any quadratic form $f$ can be written as $$\sum_{i,j=1}^{n} a_{i,j}x_i x_j,$$ where $a_{i,j}=a_{j,i}$. The symmetric matrix $A=(a_{ij})$ is called the matrix of the quadratic form $f$.
If we let $X$ denote the column vector of the variables $x_1,\dots,x_n$ then the quadratic form can be written as $$f = X^tAX.$$
Suppose now we replace the variables $x_1,\dots,x_n$ by the new variables $y_1,\dots,y_n$ according to the formula $$x_i = \sum_{j=1}^{n} c_{ij}y_j \quad (1 \leq i \leq n, c_{ij}\in\mathbb{K}).$$
In matrix form this linear substitution becomes $X=CY$, where $Y$ is the column vector of the variables $y_1,\dots,y_n$, and $C=(c_{ij})$.
If we replace the variables in $f$ by their corresponding expression in $y_1,\dots,y_n$ we shall obtain a quadratic form $g$. The matrix $A'$ of the quadratic form $g$ equals $$A' = C^tAC.$$
Throughout the text mentioned above, $\mathbb{K}$ denotes an arbitrary field with characteristic $\neq 2$. Now, I understand that this might be a really naïve question, but I am not able to actually understand what leads to the expression in matrix form $A' = C^tAC$. I am still a first-year undergraduate, I started actually studying linear algebra a semester ago...
This textbook was strongly recommended to me during a summer programme in number theory I took part in twice: PROMYS Europe. I am finding the book itself rather readable, and the topics covered do interest me a lot! At the same time I am realising I do not have enough background to tackle some of the supplements (which are sometimes referred to during the rest of the book). Can you suggest me some sources, either textbooks or online material, where I can find out more about quadratic forms and finite fields without taking a full algebra course?
Observe we have $\;f=X^tAX\;$ , and then we have $\;X=YC\;$ , so we get
$$f=X^tAX=(CY)^tA(CY)=Y^tC^tACY=Y^t\left(C^tAC\right)Y$$
Since $\;Y,\,Y^t\;$ are just the vector and its transpose on which the form acts, the new matrix is $\;C^tAC\;$.
Finally: I'm not completely sure, but I think this stuff may be a little advanced for a first year undergraduate. Were you given this stuff to work out, or else you alone decided to make some self study on this?