Linear Dependence and Independence

Definition

Let \( V \) be a vector space over a field \( K \).

Vectors \( v_1, v_2, \ldots, v_m \) in \( V \) are linearly dependent if there exist scalars \( a_1, a_2, \ldots, a_m \) in \( K \), not all zero, such that:

\[
a_1 v_1 + a_2 v_2 + \cdots + a_m v_m = 0
\]

Otherwise, the vectors are linearly independent.

Restated Definition

Consider the vector equation:

\[
x_1 v_1 + x_2 v_2 + \cdots + x_m v_m = 0
\]

If the only solution is \( x_1 = 0, x_2 = 0, \ldots, x_m = 0 \), then the vectors are linearly independent. Otherwise, they are linearly dependent.

Set of Vectors

• A set \( S = \{v_1, v_2, \ldots, v_m\} \) is dependent or independent based on the vectors in the set.
• An infinite set is dependent if some vectors within it are dependent.

Linear Dependence in \( \mathbb{R}^3 \)

• Any two vectors in \( \mathbb{R}^3 \) are dependent if they lie on the same line through the origin.
• Any three vectors in \( \mathbb{R}^3 \) are dependent if they lie on the same plane through the origin.
• Four or more vectors in \( \mathbb{R}^3 \) are always dependent.

Example: Dependent Vectors

Let \( u = (1, 1, 0) \), \( v = (1, 3, 2) \), \( w = (4, 9, 5) \)

\[
3u + 5v – 2w = 3(1,1,0) + 5(1,3,2) – 2(4,9,5) = (0,0,0)
\]

Conclusion: Non-trivial solution exists \( \Rightarrow \) Linearly Dependent

Example: Independent Vectors

Let \( u = (1,2,3) \), \( v = (2,5,7) \), \( w = (1,3,5) \)

\[
x \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + y \begin{bmatrix} 2 \\ 5 \\ 7 \end{bmatrix} + z \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
\]

Solving gives \( x = 0, y = 0, z = 0 \) ⟹ Only trivial solution ⟹ Linearly Independent

Example: Functions

Let \( f(t) = \sin t \), \( g(t) = e^t \), \( h(t) = t^2 \)

\[
x \sin t + y e^t + z t^2 = 0
\]

Testing at \( t = 0, \pi, \frac{\pi}{2} \) gives \( x = 0, y = 0, z = 0 \)

Conclusion: Only trivial solution ⟹ Linearly Independent

Key Remarks

• If zero vector is included ⟹ Linearly Dependent
• A single non-zero vector ⟹ Linearly Independent
• Scalar multiples ⟹ Linearly Dependent
• Order does not matter for independence
• Subsets of independent sets are independent

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Linear Dependence and Independence

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