Master n-Dimensional Vectors: Linear Combinations, Independence, and Rank

n‑Dimensional Vectors

An ordered list of numbers arranged in a single row is called a row vector, while arranging them in a column yields a column vector. Both are collectively referred to as n‑dimensional vectors. The i‑th number of a vector is its i‑th component. Vectors with real components are real vectors; those with complex components are complex vectors.

Zero Vector

A vector whose components are all zero is the zero vector, denoted 0.

Vector Group

A set of column vectors (or row vectors) of the same dimension forms a vector group. For example, a matrix with m rows consists of m column vectors, which together constitute the matrix’s column vector group. The matrix also has n row vectors.

Linear Combination

Given a vector group \(\{v_1, v_2, \dots, v_k\}\) and any set of scalars \(\{a_1, a_2, \dots, a_k\}\), the vector \(a_1v_1 + a_2v_2 + \dots + a_kv_k\) is called a linear combination of the group, and the scalars \(a_i\) are its coefficients.

Linear Representation

For a given vector group \(\{v_1, \dots, v_k\}\) and a vector \(u\), if there exist scalars \(c_1, \dots, c_k\) such that \(u = c_1v_1 + \dots + c_kv_k\), then \(u\) can be linearly represented by the group.

Theorem 1

A vector \(u\) can be linearly represented by a vector group \(V\) if and only if the rank of the matrix formed by appending \(u\) to the columns of \(V\) equals the rank of the matrix formed by \(V\) alone.

Linear Dependence

A set of n‑dimensional vectors \(\{v_1, \dots, v_m\}\) is linearly dependent if there exists a non‑trivial set of scalars, not all zero, such that \(a_1v_1 + \dots + a_mv_m = 0\). If the only solution is the trivial one, the set is linearly independent.

Theorem 2

The vector group is linearly dependent if and only if the rank of its associated matrix is less than the number of vectors; it is linearly independent if and only if the rank equals the number of vectors.

Theorem 3

If a subset of a vector group is linearly dependent, then the whole group is linearly dependent; conversely, if the group is linearly independent, every subset is also linearly independent.

Maximum Linearly Independent Vector Group

Given a vector group \(A\), if one can select a subset of vectors that (1) is linearly independent and (2) any larger subset becomes linearly dependent, this subset is called a maximal linearly independent group (or maximal independent set). The number of vectors in such a set is the rank of the original group. A group consisting only of the zero vector has rank 0 and no maximal independent set.

Theorem 4

The rank of a matrix equals the rank of its column vector group, which also equals the rank of its row vector group.

Theorem 5

If a vector group \(B\) can be linearly represented by another group \(A\), then the rank of \(B\) does not exceed the rank of \(A\).

Corollary 1: Equivalent vector groups have equal rank.

Corollary 2: (statement omitted in source).

Corollary 3: If a vector group \(C\) is a subset of \(A\), \(C\) is linearly independent, and \(A\) can be linearly represented by \(C\), then \(C\) is a maximal independent set of \(A\).