Consider some linearly independent vectors v⃗1, v⃗2, …v⃗m in ℝ^n and a vector v⃗ in ℝ^n that is

Name: Problem 39, Chapter 3: Linear Algebra With Applications
Uploaded: 2023-02-09T20:44:54-08:00
Channel: Victor Salazar
Description: Consider some linearly independent vectors v⃗1, v⃗2, …v⃗m in ℝ^n and a vector v⃗ in ℝ^n that is not contained in the span of v⃗1, v⃗2, …, v⃗m . Are the vectors v⃗1, v⃗2, …v⃗m, v⃗ necessarily linearly independent? Justify your answer.

step 1 For this exercise, we have x sub 1 vector v sub 1 plus dot dot dot plus x sub m vector v sub m t

Question

Consider some linearly independent vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots$ $\vec{v}_{m}$ in $\mathbb{R}^{n}$ and a vector $\vec{v}$ in $\mathbb{R}^{n}$ that is not contained in the span of $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m} .$ Are the vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots$ $\vec{v}_{m}, \vec{v}$ necessarily linearly independent? Justify your answer.

   Consider some linearly independent vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots$ $\vec{v}_{m}$ in $\mathbb{R}^{n}$ and a vector $\vec{v}$ in $\mathbb{R}^{n}$ that is not contained in the span of $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m} .$ Are the vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots$
$\vec{v}_{m}, \vec{v}$ necessarily linearly independent? Justify your answer.

Linear Algebra With Applications

Otto Bretscher 5th Edition

Chapter 3, Problem 39

Instant Answer

Solved by Expert Victor Salazar

02/09/2023

Step 1

This means that there are no scalars $x_{1}, x_{2}, \ldots, x_{m}$ such that $x_{1}\vec{v}_{1} + x_{2}\vec{v}_{2} + \ldots + x_{m}\vec{v}_{m} = \vec{v}$. Show more…

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Consider some linearly independent vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots$ $\vec{v}_{m}$ in $\mathbb{R}^{n}$ and a vector $\vec{v}$ in $\mathbb{R}^{n}$ that is not contained in the span of $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m} .$ Are the vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots$ $\vec{v}_{m}, \vec{v}$ necessarily linearly independent? Justify your answer.

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Key Concepts

Augmentation of a Linearly Independent Set

Augmentation refers to the process of adding vectors to a linearly independent set. The key result is that if a new vector is not contained in the span of the existing independent vectors, then its inclusion maintains the set's linear independence, thereby extending the basis of the vector space.

Span

The span of a set of vectors is the collection of all linear combinations of those vectors. Understanding the span is crucial when determining the contribution of an additional vector; if the new vector lies outside this span, it introduces a new dimension to the space.

Linear Independence

Linear independence is a property of a set of vectors where no vector can be written as a linear combination of the others. This is fundamental in determining whether adding a new vector to a set changes its independence: if the new vector is not expressible as a combination of the existing ones, then the set remains linearly independent.

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