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Verifying Linear Independence

Verifying Linear Independence

Learn to verify if given vectors are linearly independent, and if not, find the relationship between the dependent vectors.

Linear dependence

We defined linear dependence as a collection with linearly dependent vectors if, given a collection of vectors x1,x2,,xn\bold x_1, \bold x_2, \cdots, \bold x_n, there exists a combination of scalars w1,w2,,wnw_1,w_2,\cdots,w_n with at least one nonzero value such that

w1x1+w2x2+...+wnxn=0w_1\bold{x_1}+w_2\bold{x_2}+...+w_n\bold{x_n}=\bold{0}

There could be infinitely many possible linear combinations, and it’s not possible to check them one by one. We’ve already discussed a few special cases and tricks to check for linear dependence in a combination. This lesson will use elimination as a formal method to solve the vector equation and find the possible combinations.

Procedure

Our goal is to find the combination of scalars w1,w2,,wnw_1,w_2,\cdots,w_n, such that

w1x1+w2x2+...+wnxn=0w_1\bold{x_1}+w_2\bold{x_2}+...+w_n\bold{x_n}=\bold{0}

Expanding the vectors, we get

w1[x11x12x1m]+w2[x21x22x2m]++wn[xn1xn2x1m]=[000]w_1 \begin{bmatrix} x_{11} \\ x_{12} \\ \vdots \\ x_{1m} \end{bmatrix} + w_2 \begin{bmatrix} x_{21} \\ x_{22} \\ \vdots \\ x_{2m} \end{bmatrix} + \cdots + w_n \begin{bmatrix} x_{n1} \\ x_{n2} \\ \vdots \\ x_{1m} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} ...