Solution Found!
Let x1, x2, and x3 be linearly independent vectors in Rn and let y1 = x1 + x2, y2 = x2 +
Chapter 3, Problem 6(choose chapter or problem)
Let \(\mathbf{x}_{1}, \mathbf{x}_{2}\), and \(\mathbf{x}_{3}\) be linearly independent vectors in \(\mathbb{R}^{n}\) and let
\(\mathbf{y}_{1}=\mathbf{x}_{1}+\mathbf{x}_{2}, \quad \mathbf{y}_{2}=\mathbf{x}_{2}+\mathbf{x}_{3}, \quad \mathbf{y}_{3}=\mathbf{x}_{3}+\mathbf{x}_{1}\)
Are \(\mathbf{y}_{1}, \mathbf{y}_{2}\), and \(\mathbf{y}_{3}\) linearly independent? Prove your answer.
Questions & Answers
QUESTION:
Let \(\mathbf{x}_{1}, \mathbf{x}_{2}\), and \(\mathbf{x}_{3}\) be linearly independent vectors in \(\mathbb{R}^{n}\) and let
\(\mathbf{y}_{1}=\mathbf{x}_{1}+\mathbf{x}_{2}, \quad \mathbf{y}_{2}=\mathbf{x}_{2}+\mathbf{x}_{3}, \quad \mathbf{y}_{3}=\mathbf{x}_{3}+\mathbf{x}_{1}\)
Are \(\mathbf{y}_{1}, \mathbf{y}_{2}\), and \(\mathbf{y}_{3}\) linearly independent? Prove your answer.
ANSWER:Step 1 of 2
To prove that, \(\mathbf{y}_{1}, \mathbf{y}_{2}\), and \(\mathbf{y}_{3}\) are linearly independent, check when does a linear combination of \(\mathbf{y}_{1}, \mathbf{y}_{2}\), and \(\mathbf{y}_{3}\) vanish.
Let,
\(\alpha \mathbf{y}_{1}+\beta \mathbf{y}_{2}+\gamma \mathbf{y}_{3}=0\)
Substitute the values of \(\mathbf{y}_{1}, \mathbf{y}_{2}\), and \(\mathbf{y}_{3}\) in the above equation.
\(\begin{array}{r}
\alpha\left(\mathbf{x}_{1}+\mathbf{x}_{2}\right)+\beta\left(\mathbf{x}_{2}+\mathbf{x}_{3}\right)+\gamma\left(\mathbf{x}_{3}+\mathbf{x}_{1}\right)=0 \\
(\alpha+\gamma) \mathbf{x}_{1}+(\alpha+\beta) \mathbf{x}_{2}+(\beta+\gamma) \mathbf{x}_{3}=0
\end{array}\)