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Let $ A = \begin{pmatrix} 1 & 4 & -2 & 2 \\ 0 & 1 & 2 & -5 \\ 2 & 6 & -8 & 14 \end{pmatrix} $.

Find a vector $ w \in \mathbb{R}^{3} $ that is not in the image of the linear transformation $ x \longmapsto Ax $.

Note: $ A $ is a $ (3 \times 4) $-matrix.

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Do you know that the image of the linear transformation is the column space of the matrix? If not, first step is to convince yourself that this is true. Next step is to work out the column space (space spanned by the columns).

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Hint: Start by trying to find a basis for the image of the transformation.

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