Question

Let \(\\{\mathbf{x}, \mathbf{y}, \mathbf{z}\\}\) be a linearly independent set in \(\mathbb{R}^{4}\). Show that \(\left\\{\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{e}_{k}\right\\}\) is a basis of \(\mathbb{R}^{4}\) for some \(\mathbf{e}_{k}\) in the standard basis \(\left\\{\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}, \mathbf{e}_{4}\right\\}\).

Short answer

The set \(\{\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{e}_k\}\) is a basis of \(\mathbb{R}^{4}\) for some \(\mathbf{e}_k\).

Step-by-step solution

Understand Basis Properties

To show that a set is a basis for \(\mathbb{R}^{4}\), we need to prove two things: the set is linearly independent, and it spans \(\mathbb{R}^{4}\). A basis in \(\mathbb{R}^{4}\) generally consists of 4 vectors.

Examine Given Set

The given set \(\{\mathbf{x}, \mathbf{y}, \mathbf{z}\}\) is linearly independent in \(\mathbb{R}^4\), thus it consists of exactly 3 vectors. We need one more vector to form a complete basis for \(\mathbb{R}^4\).

Choose Standard Basis Vector

Select a vector \(\mathbf{e}_k\) from the standard basis in \(\mathbb{R}^4\), which has 4 vectors: \(\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, \mathbf{e}_4\}\). Each \(\mathbf{e}_i\) is a vector with a 1 in the i-th position and 0 elsewhere.

Evaluate Linear Independence

To demonstrate linear independence for \(\{\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{e}_k\}\), assume a linear combination of these vectors equals zero: \(a\mathbf{x} + b\mathbf{y} + c\mathbf{z} + d\mathbf{e}_k = \mathbf{0}\). Since \(\{\mathbf{x}, \mathbf{y}, \mathbf{z}\}\) is independent, \(a, b, c\) must all be zero. Because \(\mathbf{e}_k\) is independent of \(\mathbf{x}, \mathbf{y}, \mathbf{z}\), \(d\) must also be zero. Thus, \(\{\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{e}_k\}\) is independent.

Confirm Spanning Property

Since we have 4 linearly independent vectors in \(\mathbb{R}^{4}\), these vectors automatically span the space. The definition of a spanning set requires that every vector in \(\mathbb{R}^4\) be expressible as a combination of these basic vectors.

Conclusion

By verifying linear independence and spanning, we conclude that \(\{\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{e}_k\}\) forms a basis in \(\mathbb{R}^{4}\) for at least one \(\mathbf{e}_k\) from the standard basis.

Key concepts

Linear Independence

Linear independence is an essential concept in linear algebra used to describe a set of vectors. Vectors are said to be linearly independent if no vector in the set can be written as a combination of the others. For example, if you have a set of vectors like \(\{\mathbf{x}, \mathbf{y}, \mathbf{z}\}\), they are linearly independent if the equation \(a\mathbf{x} + b\mathbf{y} + c\mathbf{z} = \mathbf{0}\) only has the trivial solution, where all coefficients \(a, b,\) and \(c\) are zero.

• Linearly independent vectors maintain their unique directions in space.
• They are crucial for forming a basis of a vector space because they ensure no redundancy.

In contrast, if a set is linearly dependent, some vectors in the group can be derived from a combination of others.
These concepts form the bedrock of understanding vector spaces and determining their dimensionality.

Spanning Set

A spanning set in linear algebra is a collection of vectors that can be linearly combined to fill a vector space. Specifically, if you can express every vector in the vector space as a linear combination of a set \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\), then this set spans the space. This means you can create any vector in the space using the vectors from the set and scalars (real numbers).

• A spanning set can define the size and dimensionality of the vector space.
• If the set spans the space, any vector in that space is achievable with these vectors.

In our solution, because we eventually found four linearly independent vectors in \(\mathbb{R}^4\), they span the entire space \(\mathbb{R}^4\). This ensures that our selection of standard basis vectors complements the original linearly independent set to create a basis.

Standard Basis

The standard basis in any vector space consists of unit vectors that simplify the components of more complex vectors in terms of those units. For \(\mathbb{R}^4\), the standard basis is typically \(\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, \mathbf{e}_4\}\), where each \(\mathbf{e}_i\) has a 1 in the position corresponding to its index, and 0 elsewhere. For example:

• \(\mathbf{e}_1 = (1, 0, 0, 0)\)
• \(\mathbf{e}_2 = (0, 1, 0, 0)\)

These vectors are particularly valuable because they provide a simple way to decompose any vector into a combination of them.
The standard basis in \(\mathbb{R}^4\) serves as a reference frame for creating other bases by combining with linearly independent vectors as seen in our solution, where a vector from the standard basis was added to complete the linearly independent set to form a full basis.

Vector Spaces

A vector space is a key concept in linear algebra that combines vectors with rules for vector addition and scalar multiplication. Vector spaces can have any number of dimensions, and elements, called vectors, must adhere to certain axioms like associativity, commutivity of addition, and existence of an additive identity (zero vector).

• Examples include \(\mathbb{R}^n\), the set of all \(n\)-dimensional real-valued vectors, where each dimension corresponds to a component of the vector.
• Vector spaces are used to model mathematical systems in physics, engineering, and economics.

Within any vector space, the basis provides a framework for understanding the space's structure. The solution addresses a specific task within the space \(\mathbb{R}^4\), demonstrating how a basis is constructed and what properties are necessary (like linear independence and spanning) to achieve this.