Independent Vectors

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A set of vectors $\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3, \dots$ is linearly independent, if no linear combination gives the zero vector \begin{align} c_1 \mathbf{x}_1 + c_2 \mathbf{x}_2 + c_3 \mathbf{x}_3 + \dots + c_n \mathbf{x}_n \not= \mathbf{0} \tag{1} \end{align} except for the zero combination $c_1,c_2, \dots = 0$. On the other hand a vector is called dependent, if some nonzero combination of $c_1,c_2, \dots$ gives the zero vector.

Independent.png

For example, if $\mathbf{x}_1$ is two times $\mathbf{x}_2$, then the combination \begin{align} \mathbf{x}_1 + (-2) \mathbf{x}_2 = \mathbf{0} \end{align} $c_1$ and $c_2$ are nonzero, thus $\mathbf{x}_1$ and $\mathbf{x}_2$ are said to be dependent. For three vectors in a plane

Independent2.png

one vector must be dependent on the others. If we write (1) in matrix form, and write the vectors - let's call them $\mathbf{v}_i$ - in a matrix $\mathbf{A}$ \begin{align} \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix}=\begin{bmatrix} 1 & 2 & 2.5 \\ 2 & 1 & -1 \end{bmatrix} \end{align} \begin{align} \begin{bmatrix} 1 & 2 & 2.5 \\ 2 & 1 & -1 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{align} The columns of $\mathbf{A}$ will be independent if the Nullspace of $\mathbf{A}$ is only the zero vector.


A set of vectors $\mathbf{v}_1,\mathbf{v}_2, \dots$ span a (sub)space, if all their linear combinations fill this space. For example all the linear combinations of the columns form the column space.





Video Lectures:

  • Gilbert Strang - Introduction to Linear Algebra Lec. 9