Linear Combinations in Abstract Vector Spaces

What a Linear Combination Means

In a familiar coordinate space, a linear combination is calculated by scaling vectors and adding the results. The same operation makes sense in any vector space, even when its elements are polynomials, matrices, or functions rather than coordinate columns. The key is that the coefficients come from the scalar field and the addition and scalar multiplication are the operations of the vector space.

Let \(V\) be a vector space over a field \(\mathbb{F}\). Unless specified otherwise, examples in this tutorial use \(\mathbb{R}\) as the field of scalars. If \(\mathbf{v}_1,\dots,\mathbf{v}_p\) is an ordered list of vectors in \(V\), and \(c_1,\dots,c_p\in\mathbb{F}\), then each coefficient \(c_i\) is paired with the vector \(\mathbf{v}_i\) in the same position. The number of vectors and the number of coefficients must match.

The same definition applies whether \(V\) is finite-dimensional or not. The expression is finite because it uses a finite list of vectors; the space itself need not have finite dimension. The order of the list matters for pairing coefficients correctly, although rearranging the terms together with their coefficients does not change the resulting vector. A coefficient can be zero, and a vector can appear more than once.

For instance, in the polynomial space \(P_2(\mathbb{R})\), whose elements are real polynomials of degree at most \(2\), the expression \(2p-3q+r\) is a linear combination of \(p,q,r\). In the matrix space \(\mathbb{R}^{2\times2}\), the expression \(2\mathbf{A}-\mathbf{B}\) is a linear combination of \(\mathbf{A}\) and \(\mathbf{B}\). The operations are performed using polynomial addition and scalar multiplication in the first case, and matrix addition and scalar multiplication in the second.

Coefficient Rules

For a fixed ordered list, linear combinations can be added or scaled by updating their coefficients. These rules follow from the distributive and associative laws in a vector space; they are not special to coordinate vectors.

Proof. By associativity and commutativity of vector addition, the terms in the two sums may be grouped by their matching vectors. Distributivity of scalar multiplication over vector addition then gives \(\left(\sum_{i=1}^{p}a_i\mathbf{v}_i\right)+\left(\sum_{i=1}^{p}b_i\mathbf{v}_i\right)=\sum_{i=1}^{p}(a_i\mathbf{v}_i+b_i\mathbf{v}_i)\). Distributivity of scalar addition gives \(a_i\mathbf{v}_i+b_i\mathbf{v}_i=(a_i+b_i)\mathbf{v}_i\) for each \(i\), proving the first identity. For the second identity, distributivity of scalar multiplication over a vector sum gives \(t\left(\sum_{i=1}^{p}a_i\mathbf{v}_i\right)=\sum_{i=1}^{p}t(a_i\mathbf{v}_i)\). Compatibility of scalar multiplication gives \(t(a_i\mathbf{v}_i)=(ta_i)\mathbf{v}_i\), proving the second identity. \(\square\)

In particular, a zero coefficient contributes the zero vector, and a negative coefficient means scaling by the additive inverse of that scalar. The notation does not change when the vectors belong to a less familiar space; only the way the vector operations are carried out changes.

Calculating in Polynomial and Matrix Spaces

For matrices, addition and scalar multiplication act on corresponding entries. Thus a matrix linear combination is calculated entry by entry, just as a polynomial combination is calculated coefficient by coefficient. The result remains in the same matrix space because the operations produce a matrix with the same number of rows and columns.

Coordinate Vectors as One Example

In \(\mathbb{R}^n\), vectors are columns, and addition and scalar multiplication are componentwise. Consequently, the \(i\)th component of a linear combination is the same linear combination of the \(i\)th components of the vectors. This is a useful computational rule, but it is an instance of the general definition, not a replacement for it.

Proof. For a single vector, scalar multiplication gives the \(i\)th component of \(c_j\mathbf{v}_j\) as \(c_jv_{ij}\), by the definition of scalar multiplication in \(\mathbb{R}^n\). Vector addition adds corresponding components. Adding the \(p\) scaled vectors therefore adds their \(i\)th components, giving \(\sum_{j=1}^{p}c_jv_{ij}\). This holds for every \(i\), so it determines every component of the resulting column vector. \(\square\)

Zero Coefficients and Zero Results

If every coefficient is zero, the resulting linear combination is the zero vector. The converse is not guaranteed: nonzero coefficients can cancel one another. This distinction matters whenever a calculation produces \(\mathbf{0}\); the result alone does not establish that the coefficients were all zero.

Proof. If every \(c_i=0\), then each term \(c_i\mathbf{v}_i=\mathbf{0}\), by the zero-scalar rule for vector spaces. The sum of these zero vectors is \(\mathbf{0}\). To see why the reverse implication does not hold in general, take \(\mathbf{v}_1=(1,0)^{\top}\) and \(\mathbf{v}_2=(-1,0)^{\top}\) in \(\mathbb{R}^2\). Then \(1\mathbf{v}_1+1\mathbf{v}_2=(0,0)^{\top}\), although both coefficients equal \(1\), not \(0\). \(\square\)

Likewise, different written expressions can sometimes have the same value. The definition of a linear combination specifies how to evaluate an expression; by itself, it does not assert that the coefficients producing a vector are unique. Here the task is to calculate the value of a given expression and keep its coefficient-vector pairings correct.

Common Pitfall: Mishandling a Negative Coefficient

A common error is to treat subtraction of a vector as subtraction of each of its entries without preserving any negative entries already present. For example, consider \(2(1,3)^{\top}-(4,-1)^{\top}\). A tempting but incorrect calculation is \((2,6)-(4,1)=(-2,5)\). The second component of the subtracted vector was changed from \(-1\) to \(1\), which is not allowed.

The correct calculation first scales and then subtracts the original components:

Check: the first component is \(2(1)-4=-2\), and the second is \(2(3)-(-1)=7\). The minus sign in front of the vector reverses both component signs, including the sign of its negative second component.

Before adding terms, it is useful to write each scaled vector, polynomial, or matrix explicitly. Then combine corresponding components or entries. This separates the scalar multiplication step from the addition step and makes sign errors easier to detect.