Doubt in Hoffman and Kunze Linear Algebra Section 8.5

You are making several category errors, all based on confusing abstract vectors with their coordinate representations given a certain basis.

First, $Talpha_1$ is a vector in the abstract vector space $V$. It is not necessarily a column vector in $mathbb{R}^n$. Thus your first equation $Talpha_1=Aalpha_1$ is already an error. You cannot multiply the $ntimes n$ matrix $A$ and the abstract vector $alpha_1$; the right side of your equation is simply nonsense. You are confusing the vector itself with its coordinate representation.

You can, however, multiply $A$ and the coordinate representation of $alpha_1$ in the given basis, which is just the column vector $(1, 0, ldots, 0)^T$. This is your second error: you represented $alpha_1$ using arbitrary coordinates $x_i$ when in fact the coordinates should be $x_1=1$ and $x_i=0$ otherwise, because $alpha_1$ is part of the basis. When you represent $alpha_1$ correctly in coordinates, the product is the vector $(A_{11}, 0, ldots, 0)^T$, which is indeed $A_{11}$ multiplied by the (correct) coordinate representation of $alpha_1$.

But all of this is overkill. What you need to recall is that the numbers in the first column of $A$ tell you the coordinates of the abstract vector $Talpha_1$ with respect to the given basis. (There is a theorem in the book that says this.) Since $A$ is upper triangular, the only nonzero coordinate is $A_{11}$, which tells you that the abstract vector $Talpha_1$ is just $A_{11}alpha_1$, as desired.