Linear algebra question Write explanation too if possibleCan solve using online calculators Picture of the problem is attached **Part 1: Eigenvectors and Eigenvalues**Suppose \( A \) is a \( 3 \times 3 \) matrix with the following eigenvectors and eigenvalues.\[\vec{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \text{ with eigenvalue } \lambda = 1,\]\[\vec{v}_2 = \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix}, \text{ with eigenvalue } \lambda = 0.5,\]\[\vec{v}_3 = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \text{ with eigenvalue } \lambda = 0.5,\](a) Write \( \vec{x} \) in the coordinates of the basis \(\{\vec{v}_1, \vec{v}_2, \vec{v}_3\}\).\[\vec{x} = \begin{bmatrix} 7 \\ 5 \\ 4 \end{bmatrix}\](b) Find \( A^k\vec{x} \) (in the standard coordinates) and the coordinates of \( A^k\vec{x} \) in the basis \(\{\vec{v}_1, \vec{v}_2, \vec{v}_3\}\) for \( k = 1, 2, 3, 4, 5 \).(c) Find \( \lim_{k \to \infty} A^k\vec{x} \), and compare your answer to the vectors \(\vec{v}_1, \vec{v}_2, \vec{v}_3\). Comment on why the limit is what it is.

Given that A is a 3×3 matrix with the following eigenvectors and eigenvalues.…

Answered: (a) Write i in the coordinates of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Linear algebra question

Write explanation too if possible

Can solve using online calculators

Picture of the problem is attached

**Part 1: Eigenvectors and Eigenvalues**

Suppose \( A \) is a \( 3 \times 3 \) matrix with the following eigenvectors and eigenvalues.

\[
\vec{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \text{ with eigenvalue } \lambda = 1,
\]

\[
\vec{v}_2 = \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix}, \text{ with eigenvalue } \lambda = 0.5,
\]

\[
\vec{v}_3 = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \text{ with eigenvalue } \lambda = 0.5,
\]

(a) Write \( \vec{x} \) in the coordinates of the basis \(\{\vec{v}_1, \vec{v}_2, \vec{v}_3\}\).

\[
\vec{x} = \begin{bmatrix} 7 \\ 5 \\ 4 \end{bmatrix}
\]

(b) Find \( A^k\vec{x} \) (in the standard coordinates) and the coordinates of \( A^k\vec{x} \) in the basis \(\{\vec{v}_1, \vec{v}_2, \vec{v}_3\}\) for \( k = 1, 2, 3, 4, 5 \).

(c) Find \( \lim_{k \to \infty} A^k\vec{x} \), and compare your answer to the vectors \(\vec{v}_1, \vec{v}_2, \vec{v}_3\). Comment on why the limit is what it is.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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