Understanding Vectors and Subspaces: A Comprehensive Guide

When working with vectors in three-dimensional space, it's essential to understand the properties and conditions that define subspaces and unit vectors. In this article, we will explore these concepts in detail, focusing on the given conditions and their implications.

Unit Vectors in R3

In the context of three-dimensional space, R3, a unit vector is a vector whose length is 1. The standard unit vectors in R3 are typically denoted as , , and . If a vector (v frac{a}{sqrt{3}} mathbf{i} frac{b}{sqrt{3}} mathbf{j} frac{c}{sqrt{3}} mathbf{k}) under given conditions (V) is a single-line with unit vector (pm frac{1}{sqrt{3}} mathbf{i} pm frac{1}{sqrt{3}} mathbf{j} pm frac{1}{sqrt{3}} mathbf{k}), it means that (a b c).

Conditions for Subspaces

Let us consider the set of all points (a) in R3 where (a a). This set can be viewed as a subspace if it satisfies the three conditions for a subspace:

It contains the zero vector. It is closed under vector addition. It is closed under scalar multiplication.

Subspace Analysis

To determine if the given set of points (a, a, a) in R3 forms a subspace, we need to verify these conditions:

1. Zero Vector

For the set to contain the zero vector, there must exist a point (a 0, 0, 0). Clearly, the point (0, 0, 0) is in the set, so the first condition is satisfied.

2. Vector Addition

If we take two points (a_1 a, a, a) and (a_2 a, a, a), their sum is (a_1 a_2 (a a, a a, a a)). For this to be in the set, each component must be equal, i.e., (2a a). This implies that (a 0). Hence, the set only contains the zero vector, and it is not closed under general vector addition unless (a 0).

3. Scalar Multiplication

For any scalar (c), the point (c(a, a, a)) is ((ca, ca, ca)). This point is in the set if (ca ca ca), which is always true for any (c) and (a). However, since the set does not contain any non-zero points due to the non-closure under vector addition, this condition is trivially satisfied.

Vector Span Analysis

Alternatively, we can show that the given set is the span of one vector. The set (a a, a, a) can be seen as the span of the vector (mathbf{v} (1, 1, 1)). The span of (mathbf{v}) consists of all vectors of the form (c(1, 1, 1)), which is equivalent to (c, c, c). This confirms that the set is indeed the span of (mathbf{v}).

Conclusion

In conclusion, the set of all points (a a, a, a) in R3 forms a subspace only if (a 0). This is because the set only contains the zero vector, and it is trivially closed under scalar multiplication but not under general vector addition. Furthermore, the set is the span of the vector (1, 1, 1).

Understanding these concepts is crucial for working with vectors and subspaces in higher-dimensional spaces. If you have any further questions, consider exploring resources on linear algebra for a deeper understanding.