given two basis and the matrix, how to find the linear transformation?

linear-algebra linear-transformations matrices

Solution 1:

The idea is: given $(x,y,z)$, write it as a linear combination in its basis $B$, this is, $$ (x,y,z) = \alpha (1,-1,0)+ \beta (0,1,0) + \gamma (0,0,2). $$

Find $\alpha, \beta,\gamma$ as a function of $x,y$ and $z$.Let $T$ be the linear transformation you are looking for. When you already know $\alpha, \beta,\gamma$ , then applying $T$ in the equation above, you find $$ T(x,y,z) = \alpha T(1,-1,0)+ \beta T(0,1,0) + \gamma T(0,0,2). $$ Due to $A$ matrix definition, you already have to $T(1,-1,0)= (2,0)$, $T(0,1,0) = (0,-1)$ and $T(0,0,2)=(-1,1)$.

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