Linear Algebra Suppose for some unknown \(\vec{u}, \vec{v}, \vec{w},\) and \(\vec{a},\)\[\vec{a} = 3\vec{u} + 2\vec{v} + \vec{w} \quad \text{and} \quad \vec{a} = 2\vec{u} + \vec{v} - \vec{w}.\]22.1 Could the set \(\{\vec{u}, \vec{v}, \vec{w}\}\) be linearly independent?Suppose that \[\vec{a} = \vec{u} + 6\vec{r} - \vec{s}\]is the only way to write \(\vec{a}\) using \(\vec{u}, \vec{r}, \vec{s}\).22.2 Is \(\{\vec{u}, \vec{r}, \vec{s}\}\) linearly independent?22.3 Is \(\{\vec{u}, \vec{r}\}\) linearly independent?22.4 Is \(\{\vec{u}, \vec{v}, \vec{w}, \vec{r}\}\) linearly independent?

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Answered: Suppose for some unknown u, v, w, and…

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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Linear Algebra

Suppose for some unknown \(\vec{u}, \vec{v}, \vec{w},\) and \(\vec{a},\)

\[
\vec{a} = 3\vec{u} + 2\vec{v} + \vec{w} \quad \text{and} \quad \vec{a} = 2\vec{u} + \vec{v} - \vec{w}.
\]

22.1 Could the set \(\{\vec{u}, \vec{v}, \vec{w}\}\) be linearly independent?

Suppose that

\[
\vec{a} = \vec{u} + 6\vec{r} - \vec{s}
\]

is the only way to write \(\vec{a}\) using \(\vec{u}, \vec{r}, \vec{s}\).

22.2 Is \(\{\vec{u}, \vec{r}, \vec{s}\}\) linearly independent?

22.3 Is \(\{\vec{u}, \vec{r}\}\) linearly independent?

22.4 Is \(\{\vec{u}, \vec{v}, \vec{w}, \vec{r}\}\) linearly independent?

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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