If {v1,v2,v3}{*v*1​,*v*2​,*v*3​} forms a basis of R3R3, then which of the following are true?

{v1,v2,v1+v3}{*v*1​,*v*2​,*v*1​+*v*3​} forms a basis of R3R3.

{v1,v1+v2,v1+v3}{*v*1​,*v*1​+*v*2​,*v*1​+*v*3​} forms a basis of R3R3.

{v1,v1+v2,v1−v3}{*v*1​,*v*1​+*v*2​,*v*1​−*v*3​} forms a basis of R3R3.

{v1,v1−v2,v1−v3}{*v*1​,*v*1​−*v*2​,*v*1​−*v*3​} forms a basis of R3R3.

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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If {v1,v2,v3} forms a basis of R3, then which of the following are true? {v1,v2,v1+v3} forms a basis of R3. {v1,v1+v2,v1+v3} forms a basis of R3. {v1,v1+v2,v1−v3} forms a basis of R3. {v1,v1−v2,v1−v3} forms a basis of R3.

Analyze whether different sets of vectors form a basis in R3 by examining linear independence and the number of vectors. This problem involves key concepts from linear algebra, including basis and vector spaces. Suitable for college-level students studying linear algebra.

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