Assume A ? Mn(?), and that x⃗1⃗, x⃗2⃗ ? ?n are ?-eigenvectors, for eigenva

Okay, so in this question the matrix A is given and we need to just find that whether the matrix

Hi here for the given question, we are given some statement in the question. We need to check wh

Hello students in this question let's consider the equation AX is equal to B. If A has linearly

Hi, so for this exercise we need to we have a two by two matrix a generic two by two matrix and

Okay, very well, so let's get started with part A of this exercise. Well, it is indeed true that

Question

Assume A ? Mn(?), and that \(\vec{x_1}, \vec{x_2} ? ?n\) are ?-eigenvectors, for eigenvalues ?1, ?2 ? ? respectively. A student is trying to prove that if ?1 ? ?2 then \(\vec{x_1}, \vec{x_2}\) is a linearly independent sequence in ?n, and wWrites the following: Assume we have a1, a2 ? ?, and a1\(\vec{x_1}\) + a2\(\vec{x_2}\) = \vec{0}. Multiply both sides of this equation by A: a1?1\(\vec{x_1}\) + a2?2\(\vec{x_2}\) = \vec{0}. Now subtracting ?1 times the first equation from the second equation: a2(?2 - ?1)\(\vec{x_2}\) = \vec{0}. Which of the following statements about the above argument are correct? (A) (No answer given) (B) (No answer given) (C) (No answer given) (D) (No answer given) The only way a sequence of length two can be linearly dependent is if one of the vectors is a scalar multiple of the other vector. The zero vector is never an eigenvector. ?1 times the first equation is a1?1\(\vec{x_1}\) + a2?1\(\vec{x_2}\) = \vec{0}. All three equations are vector equations, where both the left and right hand sides are elements of ?n.

          Assume A ? M<sub>n</sub>(?), and that \(\vec{x_1}, \vec{x_2} ? ?<sup>n</sup>\) are ?-eigenvectors, for eigenvalues ?<sub>1</sub>, ?<sub>2</sub> ? ? respectively. A student is trying to prove that if ?<sub>1</sub> ? ?<sub>2</sub> then \(\vec{x_1}, \vec{x_2}\) is a linearly independent sequence in ?<sup>n</sup>, and
wWrites the following: Assume we have a<sub>1</sub>, a<sub>2</sub> ? ?, and a<sub>1</sub>\(\vec{x_1}\) + a<sub>2</sub>\(\vec{x_2}\) = \vec{0}. Multiply both sides of this equation by A: a<sub>1</sub>?<sub>1</sub>\(\vec{x_1}\) + a<sub>2</sub>?<sub>2</sub>\(\vec{x_2}\) = \vec{0}. Now subtracting ?<sub>1</sub> times the first equation from the second
equation: a<sub>2</sub>(?<sub>2</sub> - ?<sub>1</sub>)\(\vec{x_2}\) = \vec{0}. Which of the following statements about the above argument are correct?
(A) (No answer given)
(B) (No answer given)
(C) (No answer given)
(D) (No answer given)
The only way a sequence of length two can be linearly dependent is if one of the vectors is a scalar multiple of the other vector.
The zero vector is never an eigenvector.
?<sub>1</sub> times the first equation is a<sub>1</sub>?<sub>1</sub>\(\vec{x_1}\) + a<sub>2</sub>?<sub>1</sub>\(\vec{x_2}\) = \vec{0}.
All three equations are vector equations, where both the left and right hand sides are elements of ?<sup>n</sup>.

Assume A ? M<sub>n</sub>(?), and that x⃗1⃗, x⃗2⃗ ? ?<sup>n</sup> are ?-eigenvectors, for eigenvalues ?<sub>1</sub>, ?<sub>2</sub> ? ? respectively. A student is trying to prove that if ?<sub>1</sub> ? ?<sub>2</sub> then x⃗1⃗, x⃗2⃗ is a linearly independent sequence in ?<sup>n</sup>, and
wWrites the following: Assume we have a<sub>1</sub>, a<sub>2</sub> ? ?, and a<sub>1</sub>x⃗1⃗ + a<sub>2</sub>x⃗2⃗ = 0⃗. Multiply both sides of this equation by A: a<sub>1</sub>?<sub>1</sub>x⃗1⃗ + a<sub>2</sub>?<sub>2</sub>x⃗2⃗ = 0⃗. Now subtracting ?<sub>1</sub> times the first equation from the second
equation: a<sub>2</sub>(?<sub>2</sub> - ?<sub>1</sub>)x⃗2⃗ = 0⃗. Which of the following statements about the above argument are correct?
(A) (No answer given)
(B) (No answer given)
(C) (No answer given)
(D) (No answer given)
The only way a sequence of length two can be linearly dependent is if one of the vectors is a scalar multiple of the other vector.
The zero vector is never an eigenvector.
?<sub>1</sub> times the first equation is a<sub>1</sub>?<sub>1</sub>x⃗1⃗ + a<sub>2</sub>?<sub>1</sub>x⃗2⃗ = 0⃗.
All three equations are vector equations, where both the left and right hand sides are elements of ?<sup>n</sup>.

Added by Barbara D.

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