Linear Independence Calculator

The calculator will determine whether the set of given vectors is linearly dependent or not, with steps shown.

Related calculator: Matrix Rank Calculator

$$$\mathbf{\vec{v_{1}}}$$$ $$$\mathbf{\vec{v_{2}}}$$$ $$$\mathbf{\vec{v_{3}}}$$$

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Your Input

Check whether the set of vectors $$$\left\{\left[\begin{array}{c}3\\1\\2\end{array}\right], \left[\begin{array}{c}-4\\6\\7\end{array}\right], \left[\begin{array}{c}2\\8\\9\end{array}\right]\right\}$$$ is linearly independent.

Solution

There are many ways to check whether the set of vectors is linearly independent. One of the ways is to find the basis of the vector set. If the dimension of the basis is less than the dimension of the set, the set is linearly dependent, otherwise it is linearly independent.

Thus, the basis is $$$\left\{\left[\begin{array}{c}3\\1\\2\end{array}\right], \left[\begin{array}{c}0\\\frac{22}{3}\\\frac{29}{3}\end{array}\right], \left[\begin{array}{c}0\\0\\-2\end{array}\right]\right\}$$$ (for steps, see basis calculator).

Its dimension (a number of vectors in it) is 3.

Since the dimension of the basis of the set equals the dimension of the set, the set is linearly independent.

Answer

The set of the vectors is linearly independent.