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Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $$\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ 0 & 0 & 0 & 0 \end{array}\right).$$ Does there exist a $3$-dimensional subspace of $\mathbb{R}^{4}$, $V$, and a linear transformation $S$ on $\mathbb{R}^{4}$ such that $S(\mathbf{v})=T(\mathbf{v})$ for all $\mathbf{v}\in V$ and $\left\Vert S(\mathbf{x})\right\Vert _{1}\le2$ for all $\mathbf{x}\in\mathbb{R}^{4}$ with $\left\Vert \mathbf{x}\right\Vert _{1}=1$?

Thanks in advance for any helpful answers.

  • 0
    Since $\|S(kx)\|_1 = \|kS(x)\|_1 = |k| \|S(x)\|_1$, I rather doubt it...2011-11-01
  • 7
    **Please** stop deleting questions.2011-11-05

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