Question about Content in "Linear Algebra and Its Applications" by David C Lay I'm confused about a portion mentioned in the chapter 5.4 of the book "Linear Algebra and Its Applications"
https://imgur.com/lnO3ZP1
Maybe I'm mixing the two up, but why is the vector x in V while the vector x relative to B in $\mathbb{R}^n$?
If the basis for V is B, wouldn't x relative to B be the one in V? and x on its own be the one in $\mathbb{R}^n$?
 A: The vector $x$ is given to be an element of $V$. What $[x]_B$ means, is that we are writing down the coordinates of $x$ in a given basis. This will lead to a tuple of numbers, which belongs to $\mathbb R^n$.
For example, if you take the vector space of all polynomials of degree less than or equal to $2$, $V = \{p(t) : \deg p \leq 2\}$, then $\mathbf x = 2t^2 + 3t+5$ is a polynomial, hence it is an element of $V$. However, when we consider the basis of $V$ given by $\mathcal B =\{1,t,t^2\}$, then $[\mathbf x]_{\mathcal B} = [5,3,2]$ is the representation of $\mathbf x$ in the basis $\mathcal B$, which is a series of numbers. This series of numbers belongs in $\mathbb R^3$.
So, $\mathbf x$ is a vector : an element of a vector space, so the nature of $\mathbf x$ is prescribed by the nature of $V$ : if $V$ contains polynomials, then $\mathbf x$ is a polynomial, for example. However, in a given basis, $\mathbf x$ can be expressed as a unique linear combination of the basis elements, and the unique coefficients together form a tuple of real numbers (or , a member of $\mathbb R^n$) which describes $\mathbf x$ in the given basis.
