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7. For each of the following parts, (a) determine whether or not the set S is a basis for the vector space V, (b) if S is a basis for V, find [u]s for the given u∈V, and (c) if S is not a basis for V, clearly state how you know S is not a basis. Note: If S is not a basis for V, you do not need to show the basis properties that S satisfies. a. $$S = \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix} \right\}$$ $$V = \mathbb{R}^3$$ $$u = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$$ b. $$S = \{1 + x, x^2 + 5 - x\}$$ $$V = \mathbb{P}_2$$ $$u(x) = 3 - x^2$$ c. $$S = \left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}$$ $$V = Span(S)$$ $$u = \begin{bmatrix} 2 \\ 5 \\ 0 \end{bmatrix}$$

          7. For each of the following parts,
(a) determine whether or not the set S is a basis for the vector space V,
(b) if S is a basis for V, find [u]s for the given u∈V, and
(c) if S is not a basis for V, clearly state how you know S is not a basis.
Note: If S is not a basis for V, you do not need to show the basis properties that S satisfies.
a. $$S = \left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix} \right\}$$  $$V = \mathbb{R}^3$$  $$u = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$$
b. $$S = \{1 + x, x^2 + 5 - x\}$$  $$V = \mathbb{P}_2$$  $$u(x) = 3 - x^2$$
c. $$S = \left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}$$  $$V = Span(S)$$  $$u = \begin{bmatrix} 2 \\ 5 \\ 0 \end{bmatrix}$$

7 for each of the following parts a determine whether or not the set s is a basis for the vector space v b if s is a basis for v find us for the given uv and c if s is not a basis for v clea 89467

Added by Rachel T.

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