Question: Vector Spaces. Let \(\mathbb{R}^{2 \times 2}\) be the set of all \(2 \times 2\) real matrices.

1. Verify that \(\mathbb{R}^{2 \times 2}\) is a vector space under usual matrix addition and scalar multiplication.
2. What is the dimension of \(\mathbb{R}^{2 \times 2}\) ?
3. Find a basis for \(\mathbb{R}^{2 \times 2}\).
4. Let

Is the set \(\left\{I, A, A^{2}\right\}\) linearly dependent or independent in \(\mathbb{R}^{2 \times 2}\) ?

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