We are tasked with determining whether each set of vectors forms a basis for $\mathbb{R}^3$. Since we are given that $\{v_1, v_2, v_3\}$ forms a basis of $\mathbb{R}^3$, we know that $v_1, v_2, v_3$ are linearly independent and span the vector space $\mathbb{R}^3$.

Key points to check:

1. A set of vectors forms a basis for $\mathbb{R}^3$ if and only if the vectors are linearly independent and there are exactly 3 vectors.
2. Linear independence means no vector in the set can be written as a linear combination of the others.

Let's examine each set of vectors to see if they form a basis.

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1. Set: $\{v_1, v_2, v_1 + v_3\}$

• Consider the linear combination: $c_1 v_1 + c_2 v_2 + c_3 (v_1 + v_3) = 0$ Expanding this equation: $c_1 v_1 + c_2 v_2 + c_3 v_1 + c_3 v_3 = 0$ Grouping like terms: $(c_1 + c_3) v_1 + c_2 v_2 + c_3 v_3 = 0$ Since $v_1, v_2, v_3$ are linearly independent, the only solution to this equation is: $c_1 + c_3 = 0, \quad c_2 = 0, \quad c_3 = 0$ From $c_1 + c_3 = 0$, we get $c_1 = 0$ and $c_3 = 0$, and since $c_2 = 0$, the vectors are linearly independent.

  Conclusion: True — The set $\{v_1, v_2, v_1 + v_3\}$ forms a basis of $\mathbb{R}^3$.

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