Let S₁, S₂ be subsets of a vector space over field F, with S₁ S₂. Prove that if S₁is linearly dependent then so is S₂. Find a basis for S=\left\{\left(\begin{array}{c} a+b+c \\ a-b-c \\ b+c \\ 2 a+b+c \end{array}\right): a, b, c \in \mathbb{R}\right\} Consider the vector space M2x2(R). Explain why \mathcal{A}=\left\{\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{cc} 0 & 1 \\ 0 & -1 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 1 & 1 \end{array}\right),\left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) \cdot\right\} does not form a basis for M2x2 (R). Find a basis, B, for the subspace of M2×2 (R)spanned by A. Extend B to a basis for M2x2 (R).

Solved: Let S₁, S₂ be subsets of a vector space over field F, with S₁ S₂. Pro

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let s s be subsets of a vector space over field f with s s prove that

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Let S₁, S₂ be subsets of a vector space over field F, with S₁ S₂. Prove that if S₁is linearly dependent then so is S₂. Find a basis for S=\left\{\left(\begin{array}{c} a+b+c \\ a-b-c \\ b+c \\ 2 a+b+c \end{array}\right): a, b, c \in \mathbb{R}\right\} Consider the vector space M2x2(R). Explain why \mathcal{A}=\left\{\