Problem 44 - Differential Equations and Linear Algebra

Question

Give a geometric description of the subspace of \(\mathbb{R}^{3}\) spanned by the given set of vectors. $$\\{0\\}$$

Step-by-Step Solution

Answer

The geometric description of the subspace of \(\mathbb{R}^3\) spanned by the given set of vectors \(\{0\}\) is a single point located at the origin (0, 0, 0) in the three-dimensional coordinate system.

Step 1: Observe the given vector set

The set of vectors given to us consists of just one vector, which is the zero vector (\(\{0\}\)). In this case, we have: $$ \{0\} = \{ (0, 0, 0) \} $$

Step 2: Consider all the linear combinations of the given vector set

The subspace spanned by the given vectors is the set of all possible linear combinations of these vectors. In this case, it is the set of all possible linear combinations of the zero vector: $$ a(0, 0, 0) = (0, 0, 0), $$ where \(a\) is any scalar.

Step 3: Determine the span of the given vector set

Since the only linear combination of the zero vector is the zero vector itself, the span of the given vector set is: $$ \text{span}\{(0, 0, 0)\} = \{(0, 0, 0)\} $$

Step 4: Provide a geometric description of the subspace

The subspace of \(\mathbb{R}^3\) spanned by the given set of vectors is just the origin (0, 0, 0). Geometrically, this is a single point located at the origin of the three-dimensional coordinate system.

Key Concepts

Subspace Defined by Vectors

When we talk about a subspace defined by vectors in a vector space such as \( \mathbb{R}^3 \) (the space of all three-dimensional vectors), we're referring to a subset of that space that is also a vector space on its own. Subspaces are closed under vector addition and scalar multiplication which means any linear combination of vectors in the subspace is also a part of the subspace.

In the context of the exercise, the subspace defined by the set \( \{0\} \) is the simplest subspace possible, consisting of just the zero vector, \( (0, 0, 0) \) in \( \mathbb{R}^3 \) space. Despite being not particularly interesting, it's indeed a valid subspace as it meets both requirements: any scalar multiple or addition of the zero vector is still the zero vector. Hence, this subspace is closed under both operations.

Linear Combinations in Vector Spaces

Linear combinations are at the heart of understanding vector spaces and their subspaces. Formally, a linear combination of a set of vectors is made by multiplying each vector by a scalar and then adding the results together.

In equation form, for vectors \( v_1, v_2, ..., v_n \) and scalars \( a_1, a_2, ..., a_n \) the linear combination would look like this: \( a_1v_1 + a_2v_2 + ... + a_nv_n \).

The exercise demonstrates a peculiar case where our set is \( \{0\} \) and any scalar \( a \) multiplied yields \( a(0, 0, 0) = (0, 0, 0) \), implying every linear combination will result in the zero vector again. This is why for the set \( \{0\} \) in \( \mathbb{R}^3 \) the only linear combination is the vector itself, which constitutes the entirety of the subspace.

Geometric Interpretation of Vector Span

The span of a set of vectors can be seen as the collection of all possible points you can reach by walking in directions specified by the vectors and by scaling those vectors as necessary. Geometrically, the span signifies the dimensions of the space occupied by all linear combinations of the vectors.

In our exercise, the span of \( \{0\} \) results in no 'walk' so to say, as the only available direction is staying put at the zero vector \( (0, 0, 0) \). Geometrically, this means the subspace is a single point at the origin of \( \mathbb{R}^3 \) with no length, width, or height. When more vectors are included, and uniquely so, the span can form lines, planes, or the whole of \( \mathbb{R}^3 \) depending on if the vectors are linearly dependent or independent and the number of vectors involved.