Math Problem Statement
Determine if the vectors v1 = (1, 1, 1), v2 = (2, 2, 0), v3 = (3, 0, 0) span R^3. Do this by computing the determinant of the coefficient matrix for (x, y, z) = av1 + bv2 + cv3.
Solution
To determine if the vectors , , and span , we need to compute the determinant of the coefficient matrix formed by these vectors.
The coefficient matrix, with each vector as a column, is:
1 & 2 & 3 \\ 1 & 2 & 0 \\ 1 & 0 & 0 \\ \end{pmatrix}$$ Now, let's calculate the determinant of this matrix: 1. Expand along the first row: $$\text{det} = 1 \cdot \begin{vmatrix} 2 & 0 \\ 0 & 0 \end{vmatrix} - 2 \cdot \begin{vmatrix} 1 & 0 \\ 1 & 0 \end{vmatrix} + 3 \cdot \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix}$$ 2. Calculate each minor: - $$\begin{vmatrix} 2 & 0 \\ 0 & 0 \end{vmatrix} = (2 \cdot 0) - (0 \cdot 0) = 0$$ - $$\begin{vmatrix} 1 & 0 \\ 1 & 0 \end{vmatrix} = (1 \cdot 0) - (1 \cdot 0) = 0$$ - $$\begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} = (1 \cdot 1) - (1 \cdot 2) = 1 - 2 = -1$$ 3. Substitute back: $$\text{det} = 1 \cdot 0 - 2 \cdot 0 + 3 \cdot (-1) = 0 - 0 - 3 = -3$$ Since the determinant is $$-3$$, which is non-zero, the vectors $$v_1$$, $$v_2$$, and $$v_3$$ span $$\mathbb{R}^3$$. The answer is not provided in the options given, but the correct determinant value should be $$-3$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrix Theory
Formulas
Determinant of a 3x3 matrix
Theorems
Span and Linear Independence in R^3
Suitable Grade Level
College Level
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