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Linear Independence Calculator

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Example
Created on 2024-06-20Asked by Levi Sanchez (Solvelet student)
Determine if the vectors v1=(123) \mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} , v2=(456) \mathbf{v}_2 = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} , and v3=(789) \mathbf{v}_3 = \begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix} are linearly independent.

Solution

To determine if the vectors v1=(123) \mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} , v2=(456) \mathbf{v}_2 = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} , and v3=(789) \mathbf{v}_3 = \begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix} are linearly independent, we form the matrix A A with these vectors as columns and calculate its determinant: A=(147258369) A = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix} The determinant of A A is given by: det(A)=1∣5869∣−4∣2369∣+7∣2358∣ \text{det}(A) = 1 \begin{vmatrix} 5 & 8 \\ 6 & 9 \end{vmatrix} - 4 \begin{vmatrix} 2 & 3 \\ 6 & 9 \end{vmatrix} + 7 \begin{vmatrix} 2 & 3 \\ 5 & 8 \end{vmatrix} Calculate each 2x2 determinant: ∣5869∣=(5⋅9)−(8⋅6)=45−48=−3 \begin{vmatrix} 5 & 8 \\ 6 & 9 \end{vmatrix} = (5 \cdot 9) - (8 \cdot 6) = 45 - 48 = -3 ∣2369∣=(2⋅9)−(3⋅6)=18−18=0 \begin{vmatrix} 2 & 3 \\ 6 & 9 \end{vmatrix} = (2 \cdot 9) - (3 \cdot 6) = 18 - 18 = 0 ∣2358∣=(2⋅8)−(3⋅5)=16−15=1 \begin{vmatrix} 2 & 3 \\ 5 & 8 \end{vmatrix} = (2 \cdot 8) - (3 \cdot 5) = 16 - 15 = 1 Substitute these values back into the determinant formula: det(A)=1(−3)−4(0)+7(1)=−3+7=4 \text{det}(A) = 1(-3) - 4(0) + 7(1) = -3 + 7 = 4 Since det(A)≠0 \text{det}(A) \neq 0 , the vectors v1 \mathbf{v}_1 , v2 \mathbf{v}_2 , and v3 \mathbf{v}_3 are linearly independent. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Oliver Lewis on Solvelet
1. Determine whether the vectors v1=[1,2,3] \mathbf{v}_1 = [1, 2, 3] , v2=[2,4,6] \mathbf{v}_2 = [2, 4, 6] , and v3=[3,6,9] \mathbf{v}_3 = [3, 6, 9] are linearly independent.2. Find a basis for the subspace spanned by the vectors v1=[1,0,−1] \mathbf{v}_1 = [1, 0, -1] , v2=[2,1,3] \mathbf{v}_2 = [2, 1, 3] , and v3=[0,−1,1] \mathbf{v}_3 = [0, -1, 1]
DefinitionIn a vector space, a set of vectors is said to be linearly independent if none of them can be gotten from others. A collection of vectors, { v_1, v_2,..., v_n }, is said to be linearly independent if the only way c_1 v_1 + c_2 v_2 +... + c_n v_n = 0, is if c_1 =c_2 =· · · =c_n = 0. For instance, within R^2, the vectors (1, 0) and (0, 1) are linearly independent vector spaces.
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