(Solution) - Is 1 1 1 0 1 1 0 0 1 -(2025 Original AI-Free Solution)

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Is {(1, 1, 1, (0, 1, 1), (0, 0, 1)} a basis for R3? Is {(1, 0, 0), (0, 1, 0), (0, 0, 1)}?
A linear space which has a basis with a finite number of elements is said to be finite dimensional. Otherwise, the linear space is called infinite dimensional. In a finite-dimensional space X, every basis has the same number of elements, which is called the dimension of X and denoted dim X.