Spanning $mathbb{R}^3$ with Two Vectors: Exploring Linear Independence and Cross Products

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Understanding the concept of vector spaces, particularly in the context of $mathbb{R}^3$, is fundamental in linear algebra. This article delves into how two linearly independent vectors can be used to span the entire space of $mathbb{R}^3$ through the application of linear independence and the cross product operation.

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Linear Independence and Spanning

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In linear algebra, a set of vectors is said to be linearly independent if no vector in the set can be expressed as a linear combination of the others. Mathematically, given vectors $mathbf{v_1}, mathbf{v_2}, ldots, mathbf{v_n}$, they are linearly independent if the only solution to the equation

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[c_1mathbf{v_1} c_2mathbf{v_2} ldots c_nmathbf{v_n} mathbf{0}]

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is when all coefficients $c_1, c_2, ldots, c_n$ are zero. When vectors are linearly independent, they can be combined to generate any vector in their span. In the context of $mathbb{R}^3$, if two vectors are linearly independent, they can be expanded to span the entire space.

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Introduction to Cross Product and Span

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The cross product of two vectors in three-dimensional space is a vector that is perpendicular to both of the input vectors, and its length is equal to the area of the parallelogram formed by the vectors. Notably, the cross product of two vectors does not lie within the plane spanned by those vectors, making it a useful tool in extending the span to the full space of $mathbb{R}^3$.

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Given two linearly independent vectors $mathbf{v_1} (v_{1x}, v_{1y}, v_{1z})$ and $mathbf{v_2} (v_{2x}, v_{2y}, v_{2z})$ in $mathbb{R}^3$, their cross product $mathbf{v_1} times mathbf{v_2}$ can be defined as:

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[mathbf{v_1} times mathbf{v_2} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} v_{1x} v_{1y} v_{1z} v_{2x} v_{2y} v_{2z} end{vmatrix}] (v_{1y}v_{2z} - v_{1z}v_{2y}, v_{1z}v_{2x} - v_{1x}v_{2z}, v_{1x}v_{2y} - v_{1y}v_{2x})]

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The resulting vector is orthogonal to both $mathbf{v_1}$ and $mathbf{v_2}$ and hence guarantees that the union of the vectors $mathbf{v_1}$, $mathbf{v_2}$, and their cross product $mathbf{v_1} times mathbf{v_2}$ spans $mathbb{R}^3$. This is because any vector in $mathbb{R}^3$ can be expressed as a linear combination of these three orthogonal vectors.

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Practical Application of Spanning

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Suppose we are given two vectors $mathbf{v_1} (1, 2, 3)$ and $mathbf{v_2} (4, 5, 6)$. To find a vector $mathbf{v_3}$ that is linearly independent of both $mathbf{v_1}$ and $mathbf{v_2}$, we can calculate their cross product:

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[mathbf{v_1} times mathbf{v_2} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} 1 2 3 4 5 6 end{vmatrix}] (2 cdot 6 - 3 cdot 5, 3 cdot 4 - 1 cdot 6, 1 cdot 5 - 2 cdot 4) (-3, 6, -3)]

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The vector $mathbf{v_3} (-3, 6, -3)$ is orthogonal to both $mathbf{v_1}$ and $mathbf{v_2}$, and thus, along with $mathbf{v_1}$ and $mathbf{v_2}$, spans $mathbb{R}^3$. This setup can be used in various applications, such as image processing, robotics, and 3D modeling, where the ability to span the entire space is crucial.

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Conclusion

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The concept of spanning $mathbb{R}^3$ with two vectors is a fundamental aspect of linear algebra. By understanding the properties of linear independence and the cross product, we can extend the span of any two linearly independent vectors in $mathbb{R}^3$ to cover the entire 3-dimensional space. This knowledge is invaluable in numerous fields, from computer graphics to physics, where a thorough grasp of vector operations is essential.

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For further exploration and practical application of these concepts, you may refer to advanced textbooks on linear algebra or comprehensive online resources dedicated to the topic.