What is Linear independence?

A set $S$ of vectors $v_1, v_2, ...., v_n$ are said to be linearly independent if none of the vectors in the set $S$ can be represented as a linear combination of the other vectors.

This can be represented in mathematical terms as follows:

If a trivial solutionA trivial solution is the zero solution, where all constants equal zero. is obtained to the vector equation above, the set $S$ is said to contain linearly independent vectors. Otherwise, if any $c_i$'s obtains a nonzero value, the vectors are claimed to be linearly dependent.

Note: Linearly dependent vectors can be represented as a combination of other vectors in the set.

Example 1

Determine if the vectors $\{v_1, v_2\}$ are linearly independent or not:

Find the values of $c_1$ and $c_2$ in the following vector equation:

Solve the matrix below:

The solution obtained is as follows:

Since a nontrivial was not obtained, the vectors $\{v_1, v_2\}$ are claimed to be linearly dependent.

Note: The linearly dependent vector in the set can be represented by a linear combination of other vectors in the set.

For better understanding, example 1 has been depicted graphically as below:

Example 2

Determine if the vectors $\{v_1, v_2\}$ are linearly independent or not:

Find the values of $c_1$ and $c_2$ in the vector equation:

Solve the matrix below:

The solution obtained is as follows:

A trivial solution identifies the vectors $\{v_1, v_2\}$ as linearly independent.

Note: Linearly independent vectors cannot be represented in the span of other vectors in the set. In fact, linearly independent vectors increase the span of the set.

For better understanding, example 2 has been depicted graphically as below:

Example 3

Determine if the vectors $\{v_1, v_2, v_3\}$ are linearly independent or not:

Find the values of $c_1$ and $c_2$ in the vector equation:

Solve the matrix below:

The solution obtained is as follows:

The set of vectors $\{v_1, v_2, v_3\}$ is linearly dependent. You can have two linearly independent vectors at max in $R^2$.

Note: At max $n$ linearly independent vectors can exist in a space $R^n$. Addition of another vector in the set makes it linearly dependent.

For better understanding, example 3 has been depicted graphically as below:

Quiz

Here are a few practice questions to help you understand the topic better.

Span

All the linear combinations of a vector are called its span. To check whether a point is in the span of a vector or not, use the following equation:

If any combination of $c_i$'s can generate the vector $P$, it is considered to be in the span of the set of vectors; otherwise, if all $c_i$'s equal zero, the vector $P$ is said to be out of the span of the vector set.

Linear independence and span

Linearly independent vectors increase the span of the set of vectors; however, linearly dependent vectors do not add to the span of the set vectors.

Note: To span a space of $R^n$, exactly $n$linearly independent vectors are required. A number less or greater than $n$ do not span $R^n$. The set of these $n$ vectors is also known as the basis for $R^n$.

Examples

Using the examples quoted above, determine whether the vectors span $R^2$ or not.

• Example 1: The set of vectors is linearly dependent, and removing the dependent vector leaves only one vector behind. Hence, the set of vectors does not span $R^2$.

• Example 2: The set of vectors is linearly independent, and the number of vectors is exactly 2. Hence, the set of vectors span $R^2$.

• Example 3: The set of vectors is linearly dependent; however, removing the dependent vector leaves two linearly independent vectors behind. Hence, the new set of vectors would span $R^2$ but not the existing one.