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**slevvio** $\displaystyle f: \mathbb{R}^3 \rightarrow \mathbb{R}^2 $ given by $\displaystyle f(x,y,z) = (x+ 2y + 3z, x-y) $ is a linear mapping. Determine its image and give a basis for the image.

Is this correct?

$\displaystyle Im f = \{f(\bold{x}) : \bold{x} \in \mathbb{R}^3 \} = \{f(x,y,z): x,y,z \in \mathbb{R}\} $

$\displaystyle = \{(x+2y+3z)(1,0) + (x-y)(0,1): x,y,z \in \mathbb{R} \}$

$\displaystyle = sp((1,0),(0,1)) $

So the basis is (1,0),(0,1). Any help would be appreciated.