Linear Indepedence

Instructions

Quick theory overview:

The set of vectors \(\mathbf{v_1, v_2, . . ., v_m} \in \mathbb{F}_{2^n} \) are said to be linearly dependent, if there exists a set of \(n\) scalars \(a_1, a_2, . . ., a_m \in \mathbb{F}_{2^n}\), not all zero, such that \( a_1 \mathbf{v_1} + a_2 \mathbf{v_2} + . . . + a_m\mathbf{v_m}= \mathbf{0} \).
The set of vectors \( \mathbf{v_1, v_2, \dots, v_m} \) are said to be linearly independent, if and only if the equation \( a_1 \mathbf{v_1} + a_2 \mathbf{v_2} + \dots + a_m \mathbf{v_m}= \mathbf{0} \) has \(a_1 = a_2 =\dots = a_m=0 \) as the only solution.

Procedure:

Enter the values of \(a_1, a_2, \dots, a_m \in \{0, 1\}\) in the corresponding fields.
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The correctness of the entered answer is displayed in Observations.
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For the vectors \(\mathbf{v_1}=\begin{bmatrix} 1\\ 1\\ 0\\ 0 \end{bmatrix}, \mathbf{v_2}=\begin{bmatrix} 0\\ 1\\ 0\\ 1 \end{bmatrix}\) and \(\mathbf{v_3}=\begin{bmatrix} 1\\ 0\\ 0\\ 1 \end{bmatrix}\) in \(\mathbb{F}_2 ^4\), find the values of \(a_1, a_2, a_3\) for which \(a_1 \mathbf{v_1}+a_2 \mathbf{v_2}+ a_3 \mathbf{v_3}= \mathbf{0}\) and determine whether the vectors are linearly independent or not.

Enter values in the fields below.

Observations

For the vectors \(\mathbf{v_1}=\begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix}, \mathbf{v_2}=\begin{bmatrix} 0\\ 1\\ 1\\ 1 \end{bmatrix}, \mathbf{v_3}=\begin{bmatrix} 1\\ 0\\ 1\\ 1 \end{bmatrix}\), and \(\mathbf{v_4}=\begin{bmatrix} 0\\ 1\\ 0\\ 0 \end{bmatrix}\) in \(\mathbb{F}_2 ^4\), find the values of \(a_1, a_2, a_3, a_4\) for which \(a_1 \mathbf{v_1}+a_2 \mathbf{v_2}+ a_3 \mathbf{v_3}+ a_4 \mathbf{v_4}= \mathbf{0}\) and determine whether the vectors are linearly independent or not.

Enter values in the fields below.

Observations

For the vectors \(\mathbf{v_1}=\begin{bmatrix} 1\\ 0\\ 1\\ 0 \end{bmatrix}, \mathbf{v_2}=\begin{bmatrix} 0\\ 0\\ 0\\ 1 \end{bmatrix}, \mathbf{v_3}=\begin{bmatrix} 1\\ 1\\ 1\\ 0 \end{bmatrix}\), and \(\mathbf{v_4}=\begin{bmatrix} 0\\ 0\\ 1\\ 1 \end{bmatrix}\) in \(\mathbb{F}_2 ^4\), find the values of \(a_1, a_2, a_3, a_4\) for which \(a_1 \mathbf{v_1}+a_2 \mathbf{v_2}+ a_3 \mathbf{v_3}+ a_4 \mathbf{v_4}= \mathbf{0}\) and determine whether the vectors are linearly independent or not.

Enter values in the fields below.

Observations