Question

question 18
let \\( \vec{v}_1 \\) and \\( \vec{v}_2 \\) be eigenvectors of an \\( n \times n \\) matrix \\( a \\) corresponding to eigenvalues \\( \lambda_1 = 2 \\) and \\( \lambda_2 = 5 \\), respectively. which of the following statements is always true?

(a) the set \\( \\{ \vec{v}_1, \vec{v}_2 \\} \\) is linearly independent.

(b) the set \\( \\{ \vec{v}_1, \vec{v}_2 \\} \\) is linearly dependent.

(c) \\( \vec{v}_1 \\) must be a scalar multiple of \\( \vec{v}_2 \\).

(d) the set \\( \\{ \vec{v}_1, \vec{v}_2 \\} \\) forms a basis for \\( \mathbb{r}^n \\).

(e) none of the above.

Explanation:

Step1: Recall Eigenvector Property

Eigenvectors corresponding to distinct eigenvalues of a matrix are linearly independent. Here, \(\vec{v}_1\) (eigenvalue \(\lambda_1 = 2\)) and \(\vec{v}_2\) (eigenvalue \(\lambda_2 = 5\)) have distinct eigenvalues.

Step2: Analyze Each Option

• Option A: By the property of eigenvectors (distinct eigenvalues imply linear independence), \(\{\vec{v}_1, \vec{v}_2\}\) is linearly independent.
• Option B: Contradicts the linear independence property, so false.
• Option C: Scalar multiples would mean same eigenvalue (or dependent), but here eigenvalues are distinct, so false.
• Option D: For a basis of \(\mathbb{R}^n\), we need \(n\) linearly independent vectors. Here we have 2 vectors, and \(n\) is unknown (could be more than 2), so this isn't always true.

Answer:

A. The set \(\{\vec{v}_1, \vec{v}_2\}\) is linearly independent.