1. determine whether or not the three vectors $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, $\mathbf{v}_2 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$, and $\mathbf{v}_3 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$ are linearly independent. justify your answer.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Determinant of Matrix" ], "new_concepts": [ "Linear Independence", "Vector Space Basis" ], "current_concepts": [ "Determinant of Matrix", "Linear Independence", "Vector Space Basis" ] } ## Set up the matrix of vectors We construct a matrix $A$ using the given vectors as columns: $$ A = \begin{bmatrix} \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 0 & 0 \end{bmatrix} $$ ## Compute the determinant Using the **Determinant of Matrix** knowledge point $$ \begin{aligned} \det(A) &= 1 \cdot \det\begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} - 1 \cdot \det\begin{bmatrix} 1 & 1 \ 1 & 0 \end{bmatrix} + 1 \cdot \det\begin{bmatrix} 1 & 1 \ 1 & 0 \end{bmatrix} \ &= 1(0) - 1(-1) + 1(-1) \ &= 0 + 1 - 1 \ &= 0 \end{aligned} $$ ## Analyze linear independence Since the determinant of the matrix $A$ is zero, the column vectors are linearly dependent. Alternatively, we can observe directly that $\mathbf{v}_2 = \mathbf{v}_3$, which means they are identical and thus linearly dependent. The three vectors are not linearly independent (they are linearly dependent). { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Linear Algebra", "Linear Independence" ] }

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Set up the matrix of vectors

Compute the determinant

Analyze linear independence

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