**Linear Combination and Independence in Vector Spaces**Consider a vector space \( V \) and a finite number of vectors \( \mathbf{x}_1, \ldots, \mathbf{x}_k \in V \). Then, every \( \mathbf{v} \in V \) can be expressed in the form\[\mathbf{v} = \lambda_1 \mathbf{x}_1 + \cdots + \lambda_k \mathbf{x}_k = \sum_{i=1}^{k} \lambda_i \mathbf{x}_i \in V\]with \( \lambda_1, \ldots, \lambda_k \in \mathbb{R} \) representing a linear combination of the vectors.A set of vectors is known to be linearly **independent** if no vector in the set is a linear combination of any of the other vectors. This indicates that the set of vectors has no redundancy.**Multiple Choice Options:**- ☐ dependent- ☐ independent Can the dot product of the following two matrices be computed?\[\begin{bmatrix}1 & 2 \\4 & 5 \\7 & 8 \\\end{bmatrix}\begin{bmatrix}1 & 1 & 0 \\0 & 1 & 1 \\\end{bmatrix}\]Notes: Recall the dot product in [vectors](#) and [dot product in matrices](#).- True- False

Consider a vector space V and a finite number of vectors ₁,..., xk € V. Then, every v € Vof the form v=√₁₁ + + kæk = [h=₁ \;X; ¤ V with A₁,..., A ER is a linear combination of the vectors. A set of vectors is know to be linearly ______ if no vectors in the set is a linear combination of any of the other vectors, which says that the set of vectors has no redundancy. dependent O independent

Consider a vector space V and a finite number of vectors ₁,..., xk € V. Then, every v € Vof the form v=√₁₁ + + kæk = [h=₁ \;X; ¤ V with A₁,..., A ER is a linear combination of the vectors. A set of vectors is know to be linearly ______ if no vectors in the set is a linear combination of any of the other vectors, which says that the set of vectors has no redundancy. dependent O independent

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

**Linear Combination and Independence in Vector Spaces**

Consider a vector space \( V \) and a finite number of vectors \( \mathbf{x}_1, \ldots, \mathbf{x}_k \in V \). Then, every \( \mathbf{v} \in V \) can be expressed in the form

\[
\mathbf{v} = \lambda_1 \mathbf{x}_1 + \cdots + \lambda_k \mathbf{x}_k = \sum_{i=1}^{k} \lambda_i \mathbf{x}_i \in V
\]

with \( \lambda_1, \ldots, \lambda_k \in \mathbb{R} \) representing a linear combination of the vectors.

A set of vectors is known to be linearly **independent** if no vector in the set is a linear combination of any of the other vectors. This indicates that the set of vectors has no redundancy.

**Multiple Choice Options:**
- ☐ dependent
- ☐ independent

Can the dot product of the following two matrices be computed?

\[
\begin{bmatrix}
1 & 2 \\
4 & 5 \\
7 & 8 \\
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 1 \\
\end{bmatrix}
\]

Notes: Recall the dot product in [vectors](#) and [dot product in matrices](#).

- True
- False

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