Solve this multiple choice linear algebra question, please whoe your work. **Determine if the vectors**\[ \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \text{ and }\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \]**are linearly dependent or independent. Also determine if the vectors span** \( \mathbb{R}^3 \) **or not.****Options:**- ☐ The vectors are linearly dependent and they do not span \( \mathbb{R}^3 \).- ☐ The vectors are linearly dependent and they span \( \mathbb{R}^3 \).- ☐ The vectors are linearly independent and they span \( \mathbb{R}^3 \). They form a basis for \( \mathbb{R}^3 \).- ☐ The vectors are linearly independent and they do not span \( \mathbb{R}^3 \).

Answered: Determine if the vectors 2 -2 1 " 1 2 "…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Solve this multiple choice linear algebra question, please whoe your work.

**Determine if the vectors**

\[ \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \text{ and }\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \]

**are linearly dependent or independent. Also determine if the vectors span** \( \mathbb{R}^3 \) **or not.**

**Options:**
- ☐ The vectors are linearly dependent and they do not span \( \mathbb{R}^3 \).

- ☐ The vectors are linearly dependent and they span \( \mathbb{R}^3 \).

- ☐ The vectors are linearly independent and they span \( \mathbb{R}^3 \). They form a basis for \( \mathbb{R}^3 \).

- ☐ The vectors are linearly independent and they do not span \( \mathbb{R}^3 \).

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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