# Basis of a Vector Space Examples 3

Recall from the Basis of a Vector Space that if $V$ is a finite-dimensional vector space, then a set of vectors $\{ v_1, v_2, …, v_n \}$ is said to be a basis of $V$ if $\{ v_1, v_2, …, v_n \}$ spans $V$ and $\{ v_1, v_2, …, v_n \}$ is a linearly independent set of vectors in $V$.

We will now look at some more problems regarding bases of vector spaces.

## Example 1

**Suppose that $\{ v_1, v_2, v_3, v_4 \}$ is a basis of the vector space $V$. Determine whether or not the set of vectors $\{ v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4 + v_1 \}$ is a basis of $V$ or not.**

Let $a_1, a_2, a_3, a_4 \in \mathbb{F}$ and consider the following vector equation:

(1)

Since $\{ v_1, v_2, v_3, v_4 \}$ is a basis of $V$, then $\{ v_1, v_2, v_3, v_4 \}$ is linearly independent in $V$ which implies that:

(2)

Let $a_1 = 1$. Then $a_2 = -1$, and $a_3 = 1$ and $a_4 = -1$ and so:

(3)

Therefore $\{ v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4 + v_1 \}$ is not a linearly independent set and so $\{ v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4 + v_1 \}$ is not a basis of $V$.

## Example 2

**Suppose that $\{ v_1, v_2, v_3, v_4 \}$ is a basis of the vector space $V$. Determine whether or not the set of vectors $\{ v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4 \}$ is a basis of $V$ or not.**

Let $a_1, a_2, a_3, a_4 \in \mathbb{F}$ and consider the following equation:

(4)

Since $\{ v_1, v_2, v_3, v_4 \}$ is a basis of $V$ we have that $\{ v_1, v_2, v_3, v_4 \}$ is linearly independent in $V$ and so this implies that:

(5)

Using forward substitution we have that $a_1 = a_2 = a_3 = a_4 = 0$, so $\{ v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4 \}$ is a linearly independent set of vectors in $V$. We now only need to show that this set of vectors spans $V$. This is easy to do. Since $\{ v_1, v_2, v_3, v_4 \}$ is a basis of $V$ then for scalars $b_1, b_2, b_3, b_4 \in \mathbb{F}$ we have that:

(6)

Let $a_1 = b_1$, $a_1 + a_2 = b_2$, $a_2 + a_3 = b_3$ and $a_3 + a_4 = b_4$.

Then we have that:

(7)

So indeed $\{ v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4 \}$ spans $V$ and so $\{ v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4 \}$ is a basis of $V$ as well.

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