Linear Combinations of Vectors: Expressing Target Vectors Through Scalar Multiplication

A linear combination is one of the most powerful and unifying concepts in linear algebra. Given a set of vectors v₁, v₂, ..., vₙ and scalars c₁, c₂, ..., cₙ, their linear combination is c₁v₁ + c₂v₂ + ... + cₙvₙ. This simple construction underlies everything from solving systems of equations to dimensionality reduction, Fourier analysis, and quantum mechanics.

The question "Can a target vector be expressed as a linear combination of given vectors?" is equivalent to asking "Is the target vector in the span of those vectors?" The span of a set of vectors is the set of all possible linear combinations — it represents the subspace that those vectors "reach." If the target vector lies within this subspace, a solution exists; if not, no combination of the given vectors can produce the target.

Setting Up the System

To find the coefficients, you set up a system of linear equations. If each vector has m components and you have n basis vectors, you create an augmented matrix where each column (except the last) contains one basis vector, and the last column is the target vector. This gives you an m×(n+1) augmented matrix that you solve using Gaussian elimination.

The number of equations equals the dimension of the vectors, while the number of unknowns equals the number of basis vectors. If you have exactly as many basis vectors as the dimension (n = m), the system is square and typically has a unique solution (assuming the basis vectors are linearly independent). If n > m, the system is underdetermined and may have infinitely many solutions. If n < m, the system is overdetermined and may have no solution.

Gaussian Elimination for Linear Combinations

The most reliable method for finding coefficients is Gaussian elimination. Form the augmented matrix, perform row operations to reach row-echelon form, and then back-substitute to find each coefficient. The algorithm runs in O(mn²) time and handles both square and non-square systems.

During elimination, if you encounter a row where all coefficient columns are zero but the target column is non-zero, the system is inconsistent — the target vector is not in the span. This is the mathematical way of saying "no combination of these vectors can reach that target."

Connection to Basis and Dimension

If a set of vectors spans an n-dimensional space and is linearly independent, it forms a basis. In that case, every vector in the space has a unique representation as a linear combination of the basis vectors. The coefficients in that representation are called the coordinates of the vector with respect to that basis. Changing the basis changes the coordinates but not the underlying vector — this is the principle behind coordinate transformations in computer graphics and physics.