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Thread: Linear Mapping Image

  1. #1
    Senior Member slevvio's Avatar
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    Linear Mapping Image

    $\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.
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  2. #2
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    Quote Originally Posted by slevvio View Post
    $\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.
    The image is the set, $\displaystyle \{ (x+2y+3z,x-y)|x,y,z\in \mathbb{R} \}$.
    Now, $\displaystyle (x+2y+3z,x-y) = (x,x) + (2y,-y) + (3z,0) = x(1,1)+y(1,-1)+z(1,0)$.

    Thus, $\displaystyle \{ (1,1),(1,-1),(1,0) \}$ spams the the image. But it is not a basis we just need to remove the unnecessary vectors that can be duplicated from other ones, such as, $\displaystyle (1,0) = \tfrac{1}{2} \left[ (1,1)+(1,-1) \right]$.
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