The question is specific to this document: Image Compression.
It is chapter 6 from book "A First Course in Applied Mathematics" by Jorge Rebaza. It is, a to the point explaination of DCT that is easier for an engineer to grasp. However, some things are not explained and thus here I am.
The questions are about the notation used when the book starts talking about 1D DCT; on pg.3 of the pdf it shows $x$ that is the input to be transformed. This $x$ is a vector i.e a matrix with $n$ rows and $1$ column. We then multiply this with $C$ that is $n \times n$ matrix to get a $n \times 1$ matrix $y$, that contains the DCT coefficients.
Q: Why is $x$ described fundamentally as a column matrix and not a row matrix on pg.3 of pdf? If the $x$ was row matrix i.e it was not transposed, making it $1 x n$ matrix, how would the equation to calculate $y$ change? Right now it is $y=Cx$.
Q: Why is the equation y=Cx and not written as $y'=Cx'$ where the $'$ symbol means transpose? Since we are describing both $x$ and $y$ as being transposes?
Q: When we move to 2D DCT (pg.8 in pdf) the equation is derived like this $y=C \cdot (Cx')'$ which gives $y=CxC'$. How come here we explicitly write the transpose of the operand $x$ but not in the 1D equation?
The descriptions are looking inconsistent.