# Linear Algebra: Change of Basis Matrix and standard basis

Find the matrix of transformation $\mathbb{R}^3 \to \mathbb{R}^4$ taking $v_1 = (1,2,3)^T$ to $w_1 = (1,0,0,2)^T$, $v_2 = (2,3,4)^T$ to $w_2 = (1,2,3,4)^T$ and $v_3 = (1,1,2)^T$ to $w_3 = (0,5,1,7)^T$ by first finding the change of basis matrix from the basis ${v_1,v_2,v_3}$ to the standard basis and then multiplying it by the matrix which takes the standard basis to $w_1, w_2$ and $w_3,$ respectively. Explain why this gives you the right answer.

We traditionally write vectors as column vectors. Since this takes up a lot more space on a piece of paper, we write them transposed. For example, if i write $[1,2,3]^T$ in this sentence, it takes one line, but $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ takes up a bit more than 3, which makes things hard to read and look ugly. But both denote the same vector, $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$.
• So $V = [v_1 \; v_2 \; v_3]$ and is a 3x3. But what do you do from there to get the full transformation? – King Henry V Dec 20 '15 at 18:12