Linear Independence, Spanning, and Orthogonality in ( mathbb{R}^3 )

When discussing the vector space ( mathbb{R}^3 ), the concepts of linear independence, spanning, and orthogonality play crucial roles. This article explores these concepts and provides a detailed explanation of why three linearly independent vectors in ( mathbb{R}^3 ) span the entire space. Additionally, it offers a practical demonstration of how to derive an orthogonal set from linearly independent vectors.

What Does 'Linear Independence' Mean?

Linear independence is a fundamental concept in linear algebra. Vectors ( x_1, x_2, x_3 ) in ( mathbb{R}^3 ) are said to be linearly independent if the only solution to the equation ( c_1 x_1 c_2 x_2 c_3 x_3 0 ) is ( c_1 c_2 c_3 0 ). In simpler terms, you cannot construct any of the vectors as a linear combination of the others. Geometrically, this means that the vectors are not collinear.

Spanning in ( mathbb{R}^3 )

A set of vectors spans a vector space if any vector in that space can be expressed as a linear combination of the vectors in the set. In the context of ( mathbb{R}^3 ), if ( x_1, x_2, x_3 ) are linearly independent, they form a basis for ( mathbb{R}^3 ). The dimension of ( mathbb{R}^3 ) is 3, meaning any set of 3 linearly independent vectors spans the entire space.

Formally, if ( x_1, x_2, x_3 ) are linearly independent, they form a basis of ( mathbb{R}^3 ) and span ( mathbb{R}^3 ). This is because the span of three linearly independent vectors in ( mathbb{R}^3 ) is the entire space. Therefore, any vector in ( mathbb{R}^3 ) can be written as a linear combination of ( x_1, x_2, x_3 ).

Deriving an Orthogonal Set from Linearly Independent Vectors

To further explore the relationship between linear independence and orthogonality, we can derive an orthogonal set of vectors from a given set of linearly independent vectors. This process involves removing the projection of one vector onto another.

Step-by-Step Demonstration:

Consider vectors ( x_1 ) and ( x_2 ). Let's demonstrate how to obtain an orthogonal vector to ( x_1 ) from the set ( x_1, x_2 ).

Compute the dot product ( x_1 cdot x_2 ) which equals ( |x_1| |x_2| cos theta ), where ( theta ) is the angle between ( x_1 ) and ( x_2 ). Calculate the projection of ( x_2 ) onto ( x_1 ): [ text{proj}_{x_1} x_2 frac{x_1 cdot x_2}{x_1 cdot x_1} x_1 ] Subtract this projection from ( x_2 ) to get an orthogonal vector: [ y_2 x_2 - text{proj}_{x_1} x_2 x_2 - frac{x_1 cdot x_2}{x_1 cdot x_1} x_1 ]

If ( theta ) is not zero, ( x_1 ) and ( x_2 ) are not orthogonal, and ( y_2 ) is orthogonal to ( x_1 ).

Next, we can apply the same process to ( y_2 ) and ( x_3 ) to get another orthogonal vector:

[ y_3 x_3 - text{proj}_{y_2} x_3 - text{proj}_{x_1} x_3 ]

By repeating these steps, we can ensure that the vectors ( y_1, y_2, y_3 ) are mutually orthogonal.

Conclusion

In summary, three linearly independent vectors in ( mathbb{R}^3 ) span the entire space because they form a basis. The process of orthogonalization (like the Gram-Schmidt process) allows us to transform a set of linearly independent vectors into an orthogonal set, validating their ability to span the space.

Understanding these concepts and their implications is crucial in various fields such as computer graphics, physics, and engineering. The ability to transform linearly independent vectors into an orthogonal set is particularly useful in numerical computations and optimization problems.

For more in-depth learning or to explore related topics, you can refer to advanced linear algebra textbooks or resources online.