Question

Show that every nonempty subset of an independent set of vectors is again independent.

Short answer

Every non-empty subset of an independent set is also independent because it inherits the zero combination condition.

Step-by-step solution

Understand the Definition of Linear Independence

A set of vectors \( S = \{ v_1, v_2, ..., v_n \} \) is called linearly independent if the equation \( a_1v_1 + a_2v_2 + ... + a_nv_n = 0 \) implies that all coefficients \( a_i = 0 \). We are given an independent set and need to show any subset of it maintains linear independence.

Consider a Non-Empty Subset

Let \( T \) be a non-empty subset of \( S \). Our goal is to show that \( T \) is also linearly independent. Assume \( T = \{ w_1, w_2, ..., w_m \} \), where \( \{ w_1, w_2, ..., w_m \} \subseteq \{ v_1, v_2, ..., v_n \} \).

Show Linear Independence of the Subset

To prove \( T \) is independent, assume a linear combination equals zero: \( b_1w_1 + b_2w_2 + ... + b_mw_m = 0 \). This equation states the combination of vectors in \( T \) results in the zero vector. Since \( T \subseteq S \), all vectors \( w_i \) are in \( S \), which is linearly independent.

Apply Independence of the Original Set to the Subset

Since \( b_1w_1 + b_2w_2 + ... + b_mw_m = 0 \) can be rewritten as a linear combination of vectors from \( S \), and \( S \) is independent, this can only happen if each \( b_i = 0 \). Thus, no non-trivial linear combination of vectors from \( T \) can equal zero, proving \( T \) is independent.

Key concepts

Independent Set

An independent set of vectors is an important concept in linear algebra. When we say a set of vectors is independent, we mean no vector in the set can be written as a linear combination of the others. This means you cannot use the other vectors in the set to make one of the vectors in the set.To understand if a set is linearly independent, we look at when their linear combination equals the zero vector. In mathematical terms, for a set of vectors \( \{ v_1, v_2, ..., v_n \} \), the equation \( a_1v_1 + a_2v_2 + ... + a_nv_n = 0 \) must hold only when all the coefficients \( a_i \) are zero. This guarantees that the vectors form an independent set.One interesting property of independent sets is that any non-empty subset of an independent set is also independent. This is because, if none of the vectors in the original set can be expressed in terms of the others, the same will be true for any smaller set derived from them. This feature makes it simple to work with smaller portions of data without altering independence properties.

Vectors

In mathematics, especially in linear algebra, vectors are objects that can be added together and scaled by numbers, called scalars. Normally, you'll encounter vectors as arrows that have both a direction and a magnitude. When working with them in mathematical terms, they are often seen as arrays of numbers or coordinates. Vectors are instrumental because they allow us to explore spaces, understand shapes, and even solve systems of equations. In an algebraic context, vectors can be manipulated using operations like addition, subtraction, and multiplication by a scalar. In terms of linear independence, understanding vectors means grasping that each vector must provide a direction or dimension that isn't already spanned by the others. If a vector can be formed by a combination of others, it does not contribute anything new, thus compromising the set's independence.

Linear Combination

A linear combination involves creating a new vector by adding together other vectors, each multiplied by a coefficient (which can be any real number). This concept is often used in linear algebra to combine vectors, indicating how they can form different elements of a vector space.For example, consider vectors \( \{ v_1, v_2, v_3 \} \). A linear combination might look like \( a_1v_1 + a_2v_2 + a_3v_3 \), where \( a_1, a_2, \) and \( a_3 \) are coefficients. This operation allows us to explore sets of all possible values these vectors and weights can create.Linear combinations are crucial in the context of independent sets. If a set of vectors can form the zero vector through a non-trivial linear combination, they are considered dependent. However, if only the trivial combination (all coefficients are zero) results in the zero vector, the set is independent. This relationship helps us determine which vectors contribute uniquely to the span of the entire vector space, underlining the vectors as keys to understanding the dimensions and directions of spaces.