Expressing vectors in different bases

Question

I am trying to wrap my head around the concept of expressing vectors in different bases, and how to 'translate' between them. Consider the two bases given by: $$B={e_1,e_2,e_3},\quad B^{\prime}=\left\{\left[\begin{array}{l} 1 \\0 \\0\end{array}\right],\left[\begin{array}{l}1 \\1 \\0 \end{array}\right],\left[\begin{array}{l} 1 \\1 \\1\end{array}\right]\right\}$$ If I understood correctly, the columns of $B'$ also determine the change of basis matrix: $$P=\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]$$ So that the basis vectors in $B$ can now be expressed in terms of the basis vectors in $B'$ (sorry for a bit sloppy notation): $$b_{j}=\sum_{i} p_{i j} b_{i}^{\prime}, \qquad \{b_{j}\} = B, \quad \{b_{i}^{\prime}\} = B^{\prime}$$ Which in matrix notation becomes: $$b=Pb'$$ For example, if we have a vector,$v'$, that in basis $B'$ has coordinates $(1,4,2)$, then in $B$ the same vector will have coordinates: $$v=P v^{\prime}=\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} 1 \\ 2 \\ 4 \end{array}\right]=\left[\begin{array}{l} 7 \\ 6 \\ 4 \end{array}\right]$$ Is there anything that I misunderstood or does this look correct?

Answer 1

The matrix $P$ is the matrix that changes $B'$-coordinates into $B$-coordinates. Note that $P$ arises as the matrix representantion of the identity transformation $Id:\mathbb{R}^3 \to \mathbb{R}^3$ with respect to the bases $B'$ and $B$, respectively. This is really easy to see, since the elements of $B'$ are already written in $B$-coordinates.

So, to obtain the expression in $B$-coordinates of any vector $v = a_1b'_1 + a_2b'_2 + a_3b'_3$ written in $B'$-coordinates you use multiplication by $P$

$$[v]_B = P \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix}$$ The coordinates of $[v]_B$ give you the coefficients of $v$ in $B$-coordinates. Conversely, if you wanted to obtain $B'$-coordinates from $B$-coordinates you would have to carry out the same procedure using $P^{-1}$ instead.