Math  /  Algebra

QuestionAnswer the following statements by true or false: T..) Any subset of a vector space that does not contain the zero vector is not a space. ...) The set S={fC[1,1]:f(0)=1}S=\{f \in C[-1,1]: f(0)=1\} is not a subspace of V=C[1,1]V=C[-1,1] ...) S={AR2×2:a11=0}S=\left\{A \in R^{2 \times 2}: a_{11}=0\right\} is a subspace of V=R2×2V=R^{2 \times 2} ...) S={v=(x,y)R2:x+y=1}S=\left\{v=(x, y) \in R^{2}: x+y=1\right\} is not a subspace of V=R2V=R^{2} ...) S={fC(R):f(1)=0}S=\{f \in C(R): f(1)=0\} is a subspace of V=C(R)V=C(R) ...) S={v=(x,y)R2:x+y=1}S=\left\{v=(x, y) \in R^{2}: x+y=1\right\} is not a subspace of V=R2V=R^{2}. )\ldots) is a subspace of S={v=(x,y)R2:x+y=0},V=R2S=\left\{v=(x, y) \in R^{2}: x+y=0\right\}, V=R^{2}

Studdy Solution
The set S={v=(x,y)R2:x+y=0} S = \{v=(x, y) \in R^{2}: x+y=0\} is a subspace of V=R2 V = R^{2} because it is closed under addition and scalar multiplication, and it contains the zero vector. Therefore, this statement is true.
The answers to the statements are:
1. True
2. True
3. True
4. True
5. True
6. True
7. True

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