Proving That Every Orthogonal Set of Vectors in an Inner Product Space is Linearly Independent

It is a fundamental theorem in linear algebra and functional analysis that every orthogonal set of vectors in an inner product space is linearly independent. This article will detail the proof of this theorem and explore some related concepts to provide a comprehensive understanding of the topic.

Orthogonal Vectors in an Inner Product Space

A set of vectors ({v_1, v_2, ldots, v_n}) in an inner product space is said to be orthogonal if the inner product of any two distinct vectors is zero. Mathematically, this can be expressed as:

[langle v_i, v_j rangle 0 quad forall i eq j]

Linear Independence of Vectors

A set of vectors is linearly independent if the only solution to the equation:

[c_1 v_1 c_2 v_2 ldots c_n v_n 0]

where (c_i) are scalars, is:

[c_1 c_2 ldots c_n 0]

Proof of Linear Independence for Orthogonal Vectors

To prove that an orthogonal set of vectors is linearly independent, consider the equation:

[c_1 v_1 c_2 v_2 ldots c_n v_n 0]

Take the inner product of both sides of this equation with a specific vector (v_i) (for any (i)):

[langle c_1 v_1 c_2 v_2 ldots c_n v_n, v_i rangle langle 0, v_i rangle]

Using the linearity property of the inner product, we expand the left-hand side:

[c_1 langle v_1, v_i rangle c_2 langle v_2, v_i rangle ldots c_n langle v_n, v_i rangle 0]

Since the vectors are orthogonal, we have:

[langle v_j, v_i rangle 0 quad forall j eq i]

Therefore, the equation simplifies to:

[c_i langle v_i, v_i rangle 0]

Assuming (langle v_i, v_i rangle eq 0), which is true for non-zero vectors, the only solution is:

[c_i 0]

This holds for all (i), thus proving that:

[c_1 c_2 ldots c_n 0]

Therefore, the set is linearly independent.

Conclusion

The proof above shows that any orthogonal set of vectors in an inner product space is indeed linearly independent. This is a crucial result because it establishes a fundamental property that helps in understanding the structure and behavior of vector spaces.

Additional Considerations

For clarity, it should be noted that if zero vectors are included in the set, the statement does not hold. However, for non-zero orthogonal vectors in a real or complex vector space, the result remains valid.

Furthermore, the concept of orthogonality can be more nuanced in pseudo-Riemannian spaces like Minkowski space, where the existence of a zero vector with non-zero components challenges the intuitive notion of orthogonality. In such spaces, additional considerations are required to fully understand the properties of orthogonal sets.

This article has provided a detailed proof and discussion of the linear independence of orthogonal vectors in an inner product space, along with the potential caveats and additional considerations in more complex spaces.