Linear Independence and Dependence Examples 2

Recall from the Linear Independence and Dependence page that a set of vectors $\{ v_1, v_2, ..., v_n \}$ is said to be Linearly Independent in $V$ if the vector equation $a_1v_1 + a_2v_2 + ... + a_nv_n = 0$ implies that $a_1 = a_2 = ... = a_n = 0$, that is, the zero vector is uniquely expressed as a linear combination of the vectors in $\{ v_1, v_2, ..., v_n \}$ with the coefficients all being zero.

If a set of vectors $\{ v_1, v_2, ..., v_n \}$ is not linearly independent then we say the set if Linearly Dependent and that there exists scalars $a_1, a_2, ..., a_n \in \mathbb{F}$, not all zero, such that $a_1v_1 + a_2v_2 + ... + a_nv_n = 0$.

We will now look at some more examples to regarding the linear independence / dependence of a set of vectors.

Example 1

Find a set of vectors $v_1, v_2, v_3$ (none of which are zero vectors) of the vector set $M_{22}$ such that the set $\{ v_1, v_2, v_3 \}$ is a linearly dependent set.

Recall from an earlier theorem on the Theorems Regarding Linear Independence and Dependence page that a set of vectors is linearly dependent if one vector is a scalar multiple of another. Therefore, let's construct $v_1$ to be a scalar multiple of $v_2$. Let $v_1 = \begin{bmatrix} 1 & 1\\ 1 & 1\end{bmatrix}$, $v_2 = \begin{bmatrix} 2 & 2\\ 2 & 2\end{bmatrix}$ and $v_2 = \begin{bmatrix} 1 & 2\\ 3 & 4\end{bmatrix}$. Therefore $\{ v_1, v_2, v_3 \}$ is a linearly dependent set, because the vector equation:

(1)

\begin{align} a_1\begin{bmatrix} 1 & 1\\ 1 & 1\end{bmatrix} + a_2 \begin{bmatrix} 2 & 2\\ 2 & 2\end{bmatrix} + a_3 \begin{bmatrix} 1 & 2\\ 3 & 4\end{bmatrix} = 0 \end{align}

Does not have a unique set of scalars to which this is satisfied. For example, verify that $a_1 = 2$, $a_2 = -1$, and $a_3 = 0$ satisfies the vector equation above.

Example 2

Consider the set of vectors $\{ f(x) = \frac{1}{x}, g(x) = x \}$ from the vector space of real-valued functions $F(-\infty, \infty)$. Determine if this set is linearly independent or linearly dependent.

Once again, let's look at the following vector equation:

(2)

\begin{align} a_1f(x) + a_2g(x) = 0 \\ a_1\frac{1}{x} + a_2x = 0 \\ \end{align}

We notice that the only set of scalars is $a_1 = a_2 = 0$ when $x = 1$, and so this set is linearly independent.

Example 3

Consider the set of vectors $\{ 2, 4\sin^2 x, 6\cos ^2 x \}$ from the vector space of real-valued functions $F(-\infty, \infty)$. Determine if this set is linearly independent or linearly dependent.

Let's look at the following vector equation

(3)

\begin{align} a_1 (2) + a_2(4 \sin ^2 x) + a_3 (6 \cos ^2 x) = 0 \end{align}

Notice that if $a_1 = -1$, $a_2 = \frac{1}{2}$ and $a_3 = \frac{1}{3}$ then:

(4)

\begin{align} -2 + 2 \sin ^2 x + 2 \cos ^2 x = 0 \\ -2 + 2(\sin^2 x + \cos ^2 x) = 0 \\ -2 + 2 = 0 \\ \end{align}

Therefore $\{ 2, 4\sin^2 x, 6\cos ^2 x \}$ is not a linearly independent set.

Example 4

Prove that if $\{ v_1, v_2, ..., v_n \}$ is a set of linearly independent vectors of a vector space $V$ then $\{ v_1 - v_2, v_2 - v_3, v_3 - v_4, ..., v_{n-1} - v_n, v_n \}$ is also a linearly independent set of vectors of $V$.

Since $\{ v_1, v_2, ..., v_n \}$ is a linearly independent set of vectors, it follows that the vector equation $a_1v_1 + a_2v_2 + ... + a_nv_n = 0$ is only satisfied when $a_1 = a_2 = ... = a_n = 0$. Also, the set of vectors $\{v_2, ..., v_{n-1}, v_n \}$ is also a linearly independent set, so $a_1v_2 + a_2v_3 + ... + a_{n-1}v_n \} = 0$ is only satisfied when $a_1 = a_2 = ... = a_{n-1} = 0$. Subtracting both of these equations together we get that:

(5)

\begin{align} \quad (a_1v_1 + a_2v_2 + ... + a_nv_n) - (a_1v_2 + a_2v_3 + ... + a_{n-1}v_n \}) = 0 \\ \quad a_1(v_1 - v_2) + a_2(v_2 - v_3) + ... + a_{n-1}(v_{n-1} + v_n) + a_n(v_n) = 0 \end{align}

We have already established that $a_1 = a_2 = ... = a_{n-1} = a_n = 0$, and so $\{ v_1 - v_2, v_2 - v_3, ..., v_{n-1} - v_n, v_n \}$ is a linearly independent set.