Linear Independence

Vectors $\mathbf x_1, \mathbf x_2, ... , \mathbf x_n$ are linearly independent if no linear combinations gives a zero vector $\mathbf 0$

  • $c_1 \mathbf x_1 + c_2 \mathbf x_2 + ... + c_n \mathbf x_n \ne \mathbf 0$
  • only when $\forall i: c_i = 0$ it's true that $\sum c_i \mathbf x_i = 0$


Examples

Example 1

  • suppose we have a vector $v_1$ and a vector $v_2 = c \cdot v_1$
  • 807633b501c745a595e6a0a12277cedb.png
  • this system is dependent: $v_2$ points in the same direction as $v_1$

Zero vector always means a dependence

  • suppose we have vector $v_1 \ne \mathbf 0$ and $v_2 = \mathbf 0$
  • $0 \mathbf v_1 + c \mathbf v_2 = \mathbf 0$ for any $c$


Example 2

  • suppose we have a system of two independent vectors $v_1$ and $v_2$ in $\mathbb R^2$
  • 946d4f5d4d424e449407115d672c2a69.png
  • what happens if we add 3rd vectors $v_3$?
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  • the system is no longer dependent - we always can express $v_3$ in terms of $v_1$ and $v_2$ and when we add them, we'll have 0
  • so if the number of vectors is greater than the dimensionality of these vectors, the system cannot be independent


Matrices

Columns of a matrix $A$ are independent if the Nullspace $N(A)$ contains only $\mathbf 0$

  • otherwise the columns are dependents
  • Why? recall that $N(A)$ contains the solutions to the system $A\mathbf x = \mathbf 0$
    • so there's a combination of columns with coefficients $\mathbf x$ that is equal to $\mathbf 0$
  • It's related to rank as well:
    • if $r = n$, then there are no free variables and $N(A) = \{ \, \mathbf 0 \, \}$
    • if $r < n$, there are free variables and $| N(A) | > 1$


Maximal Linearly Independent Subset

Suppose we have a set of vectors $A = \{ \mathbf a_i \}$

  • subset $A^*$ is maximal linearly independent subset of $A$ if
  • all vectors in $A^*$ are linearly independent
  • it's not contained in any other subset of linearly idependent vectors from $A$


Sources

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