Problem 4 - A Book of Abstract Algebra

Question

Prove each of the following: The set \(\\{a\\}\), containing only one nonzero vector a, is linearly independent.

Step-by-Step Solution

Answer

The set \(\{a\}\) is linearly independent because the only solution is \(c_1 = 0\).

Step 1: Understand Linear Independence

A set of vectors is linearly independent if there is no non-trivial linear combination of its vectors that results in the zero vector. This means, for each vector in the set, you cannot find coefficients (not all zero) such that the sum of these coefficients times their respective vectors equals the zero vector.

Step 2: Identify the Set and Vector

Our set is \(\{a\}\) where \(a\) is a nonzero vector. Since we have only one vector in the set, our task is to check whether this single vector is linearly independent.

Step 3: Set Up the Linear Combination Equation

For the vector \(a\), the linear combination would be \(c_1 a = 0\), where \(c_1\) is a scalar. Our goal is to find the values of \(c_1\) which satisfy this equation.

Step 4: Analyze the Equation

Since \(a\) is a nonzero vector, the equation \(c_1 a = 0\) holds true if and only if \(c_1 = 0\). Any non-zero \(c_1\) would result in a non-zero product \(c_1 a\).

Step 5: Conclude Linear Independence

Since the only solution to \(c_1 a = 0\) is \(c_1 = 0\), there are no non-trivial solutions. Thus, the set \(\{a\}\) is linearly independent by definition.

Key Concepts

Nonzero Vector

A nonzero vector is any vector in a vector space that is not equal to the zero vector. In other words, a nonzero vector has at least one non-zero component. For example, in a 3-dimensional space, a vector like \( \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \) is nonzero because its first component is not zero. Nonzero vectors are particularly important when discussing topics like linear independence because they carry information allowing for unique solutions in vector equations. Understanding whether a vector is nonzero is the stepping stone to deciphering relations between vectors in a set.

Linear Combination

A linear combination involves combining multiple vectors using scalar multiplication followed by vector addition. Suppose we have vectors \( v_1, v_2, \ldots, v_n \). A linear combination of these vectors is expressed as:

• \( c_1 v_1 + c_2 v_2 + \ldots + c_n v_n \)

This equation uses scalars \( c_1, c_2, \ldots, c_n \) as weights for each vector in the combination. The beauty of linear combinations lies in their ability to show how vectors relate to each other within a larger space. By adjusting the scalars, one can form various vectors that lay in the span of the original set. It's crucial to understand linear combinations when exploring if vectors form an independent or dependent set.

Linearly Independent Set

A linearly independent set is a collection of vectors where no vector in the set can be written as a linear combination of the others. In simpler terms, each vector adds something unique to the set. The concept is significant because it highlights when a vector set has no redundancy. For instance, consider the single vector set \( \{a\} \) where \( a \) is nonzero. To check linear independence, we use the equation:

• \( c_1 a = 0 \)

When \( c_1 \) must be zero for the equation to hold (given \( a \) is nonzero), it indicates that no non-zero linear combination results in a zero vector. Thus, \( \{a\} \) is linearly independent. Linearly independent sets are foundational in understanding vector spaces as they indicate minimal generating sets of vectors.

Zero Vector

The zero vector is unique in every vector space, being the vector where all components are zero. For example, in a 3D space, it is denoted as \( \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} \). Despite its simplicity, the zero vector is a powerful concept. It acts as the identity element in vector addition, as any vector added to the zero vector results in the original vector. In scenarios of linear dependence and independence, the zero vector has a critical property; any non-zero scalar times zero produces zero, hence any vector including the zero vector is linearly dependent. It is this behavior that provides the base checks for establishing the characteristics of vector sets.