We have $i : \mathbb{R}^2 \to \mathbb{R}^3 v \mapsto ( \langle v, w_1 \rangle , \langle v, w_2 \rangle , \langle v, w_3 \rangle )$ with $w_1 = (1,-1)$, $w_2 = (2,3)$ and $w_3 = (2,4)$. Determine the matrix $A$ such that the map is also given by $x \mapsto Ax$.

Is it then correct to say that we have $(v_1 - v_2, 2v_1 + 3v_2, -2v_1 + 4v_2)$ and hence our matrix is $$A = \begin{pmatrix}1&-1\\ 2&3\\ -2&4\end{pmatrix}$$

because $\mathbb{R^2}$ implies that $v = \begin{pmatrix} v_1\\ v_2\end{pmatrix}$?

  • $\begingroup$ Yes, You Compute right. (درست حساب کردید) $\endgroup$ – Hoseyn Heydari Oct 12 '15 at 20:28
  • $\begingroup$ To be honest: you can't unless you specify bases of $\matthbb R^2$ and $\mathbb R^3$ in respect of which $A$'s matrix representation should be given. $\endgroup$ – Michael Hoppe Oct 12 '15 at 20:31

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