Question

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).

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