Linear independence

Which of the following vectors are redundant? If there are redundant vectors, write each of them as a linear combination of previous vectors.

Use the method of Theorem 5.21 to determine whether the following vectors are linearly independent. If they are linearly dependent, find a non-trivial linear combination of the vectors that is equal to $\mathbf{0}$.

Are the following vectors linearly independent? If not, write one of them as a linear combination of the others.

Find a linearly independent set of vectors that has the same span as the given vectors.

Here are some vectors.

Describe the span of these vectors as the span of as few vectors as possible.

In this exercise, we use scalars from the field $\mathbb{Z}_{3}$ of integers modulo $3$ instead of real numbers (see Section an exercise given elsewhere in the book, “Fields”). Use the extended casting-out algorithm to determine which of the following vectors are redundant. If there are redundant vectors, write each of them as a linear combination of previous vectors.

Let $\mathbf{u},\mathbf{v},\mathbf{w}$ be linearly independent vectors in $\mathbb{R}^{n}$. Are the vectors $\mathbf{u}+\mathbf{v}$, $2\mathbf{u}+\mathbf{w}$, and $\mathbf{w}-2\mathbf{v}$ linearly independent?

Let $\mathbf{u},\mathbf{v},\mathbf{w}$ be linearly independent vectors in $\mathbb{R}^{n}$. Are the vectors $\mathbf{u}+\mathbf{v}$, $\mathbf{u}+\mathbf{w}$, and $\mathbf{w}+\mathbf{v}$ linearly independent?

Suppose $A$ is an $m\times n$-matrix and $\left\{\mathbf{w}_{1},\ldots,\mathbf{w}_{k}\right\}$ is a linearly independent set of vectors in $\mathbb{R}^{m}$. Now suppose $A\mathbf{z}_{i}=\mathbf{w}_{i}$. Show $\left\{\mathbf{z}_{1},\ldots,\mathbf{z}_{k}\right\}$ is also linearly independent.