Linearly Independent and Dependent Vectors

For vectors $v_1,v_2,\dots,v_n$:

Linearly Independent Vectors

The vectors are linearly independent if the only solution to

$$c_1v_1+c_2v_2+\cdots+c_nv_n=0$$

is

$$\boxed{c_1=c_2=\cdots=c_n=0}$$

Linearly Dependent Vectors

The vectors are linearly dependent if there is a non-zero set of coefficients satisfying

$$c_1v_1+c_2v_2+\cdots+c_nv_n=0$$

where at least one $c_i\neq0$.

Key Difference Between Linearly Independent and Dependent Vectors

So the key difference is:

$$\boxed{\text{Independent} \Rightarrow \text{only trivial solution}}$$

$$\boxed{\text{Dependent} \Rightarrow \text{a non-trivial solution exists}}$$

For example, $v_1=(1,2)$ and $v_2=(2,4)$ are dependent because

$$2v_1-v_2=0.$$

How to Check if 3 Vectors in R³ Are Linearly Independent

If you’re given 3 vectors in R³, there are two common ways to check whether they are linearly independent.

Method 1: Determinant — Fastest Method

Suppose

$$v_1=\left(\begin{array}{c}a\\ b\\ c\end{array}\right),v_2=\left(\begin{array}{c}d\\ e\\ f\end{array}\right),v_3=\left(\begin{array}{c}g\\ h\\ i\end{array}\right)$$

Put them as columns of a matrix:

$$A=\left(\begin{array}{ccc}a& d& g\\ b& e& h\\ c& f& i\end{array}\right)$$

Then calculate the determinant:

$$\det \left(A\right)$$

• If det(A) ≠ 0, the vectors are linearly independent.
• If det(A) = 0, the vectors are linearly dependent.

Method 2: Use the Linear Combination Equation

Start with

$$c_1{v}_1+c_2{v}_2+c_3{v}_3=0$$

Then solve for c₁, c₂, and c₃.

• Only solution: c₁ = c₂ = c₃ = 0 → independent.
• Any non-zero solution → dependent.

Quick Example

Consider the vectors

$$v_1=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),v_2=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),v_3=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$$

Matrix:

$$A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$

$$\det \left(A\right)=1\ne 0$$

Therefore,

$$v_1,v_2,v_3\text{ are linearly independent.}$$

Conclusion

To determine whether vectors are linearly independent or linearly dependent, you can use the linear combination equation. For 3 vectors in R³, calculating the determinant is often the fastest method: a non-zero determinant means the vectors are linearly independent, while a zero determinant means the vectors are linearly dependent.