Problem 3 - A Book of Abstract Algebra

Question

Let \(A\) be the set of eight vectors \((x, y, z)\) where \(x, y, z=1,2\). Prove that \(A\) spans \(\mathbb{R}^{3}\), and find a subset of \(A\) which is a basis of \(\mathbb{R}^{3}\).

Step-by-Step Solution

Answer

A basis for \( \mathbb{R}^{3} \) is \( \{(1,1,1), (1,1,2), (1,2,1)\} \). A spans \(\mathbb{R}^{3}\).

Step 1: Define Set A

Set \( A \) consists of all vectors of the form \( (x, y, z) \) where each component \( x, y, z \) can take the values 1 or 2. This means we need to list all such combinations of values to form the vectors in the set \( A \).

Step 2: Enumerate Vectors in A

Given the possible values for \( x, y, z \), the set \( A \) of vectors is: \( (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2) \).

Step 3: Prove A Spans \( \mathbb{R}^{3} \)

To show that \( A \) spans \( \mathbb{R}^{3} \), we need to demonstrate that any vector \( (a, b, c) \in \mathbb{R}^{3} \) can be expressed as a linear combination of the vectors in \( A \). Combining vectors, we find solutions to express the standard basis vectors \( (1,0,0), (0,1,0), (0,0,1) \) as linear combinations of vectors in \( A \).

Step 4: Identify a Basis from A

To find a basis, we need a subset of \( A \) that is both linearly independent and spans \( \mathbb{R}^{3} \). Scanning through\( A \), we choose \( (1,1,1), (1,1,2), (1,2,1) \) as a candidate basis. Check linear independence by verifying the determinant of the matrix made by these vectors is non-zero, confirming that this subset is a basis.

Step 5: Verify Linear Independence

Compute the determinant of the matrix formed by row vectors \( \begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 2 \ 1 & 2 & 1 \end{bmatrix} \). The determinant is \(1 eq 0\), confirming that this subset is linearly independent.

Key Concepts

Vectors

In Linear Algebra, vectors are fundamental building blocks. They are often represented as arrows in space that combine magnitude and direction. We can think of vectors in

• two dimensions (like points on a flat surface)
• three dimensions (like objects in our physical world)

Vectors can also be generalized to even higher dimensions. Mathematically, a vector is usually denoted in components; for example, a 3D vector is represented as \((x, y, z)\). Each component of the vector specifies the coordinate along each axis in a 3D space.
A key property of vectors is that they can be added together or multiplied by scalars (numbers) to produce another vector. This is called vector addition and scalar multiplication. When handling vectors, remember their basic operations and properties, as these form the basis for understanding deeper linear algebra concepts.

Basis

A basis in linear algebra is a set of vectors that is used to describe a space uniquely. Every vector in that space can be represented as a unique combination of the basis vectors. Think of a basis as a foundation.

• Each basis vector serves as a building block.
• The number of basis vectors determines the dimensions of the space.

For example, in three-dimensional space \(\mathbb{R}^3\), we typically consider the standard basis consisting of the vectors \((1, 0, 0)\), \((0, 1, 0)\), and \((0, 0, 1)\). Any vector in \(\mathbb{R}^3\) can be expressed as a combination of these standard basis vectors.
Finding a basis involves selecting vectors that are both linearly independent and that collectively span the entire space. In the context of the given exercise, we selected the vectors \((1,1,1)\), \((1,1,2)\), and \((1,2,1)\) to form a basis of \(\mathbb{R}^3\).

Linear Independence

Linear independence is a key concept when determining whether a set of vectors can form a basis. A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. In simpler terms, each vector adds something new,

• no redundancy
• no overlap

For a collection of vectors to be linearly independent, the only solution to the equation\[ c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + ... + c_n \mathbf{v_n} = \mathbf{0} \]is when all coefficients \(c_1, c_2, ..., c_n\) are zero.
In our example problem, the vectors \((1,1,1)\), \((1,1,2)\), and \((1,2,1)\) were checked for linear independence by computing the determinant of the matrix formed by these vectors. Because the determinant is non-zero, these vectors are indeed linearly independent.

Spanning Set

A spanning set is a collection of vectors that can be combined through addition and scalar multiplication to cover an entire vector space. If a set spans a vector space, it means that every vector in that space can be expressed as a linear combination of the vectors in the set.

• It's about coverage
• Reaching every point in the space

For instance, in three-dimensional space, having three non-parallel vectors guarantees that they can span the whole of \(\mathbb{R}^3\).
In our exercise example, the set \(A\) of vectors was shown to span \(\mathbb{R}^3\) by demonstrating that any arbitrary vector \((a, b, c)\) could be written as a linear combination of the vectors in \(A\). This confirmed that through selecting certain vectors in \(A\), the whole space \(\mathbb{R}^3\) could be represented, proving \(A\) is indeed a spanning set.