Unique Coordinates in a Fixed Basis

Why Coordinates Need an Order

A basis gives a way to describe every vector in a vector space using a finite list of building blocks. Once that list is ordered, a vector’s coefficients can be recorded in a column, so calculations with vectors can be translated into calculations with numbers. The order matters: the first entry of the coordinate column is the coefficient of the first basis vector, the second entry is the coefficient of the second, and so on. The central fact is that these coefficients are not merely one possible description. A basis guarantees that each vector has exactly one such description.

Let \(V\) be a vector space over a field \(\mathbb{F}\), and let \(\mathcal{B}=(\mathbf{b}_1,\dots,\mathbf{b}_n)\) be an ordered list of vectors in \(V\). A list is a basis if it spans \(V\) and is linearly independent. Spanning means every vector in \(V\) can be written as a linear combination of the listed vectors. Linear independence means the equation \(c_1\mathbf{b}_1+\cdots+c_n\mathbf{b}_n=\mathbf{0}\) forces every coefficient \(c_i\) to be zero. The field \(\mathbb{F}\) supplies the scalars in these combinations; the examples below use \(\mathbb{R}\).

The definition refers to a unique coefficient list. Spanning supplies at least one list, and independence rules out two different lists. These are logically separate jobs: a list that spans might allow multiple representations, while an independent list might fail to represent some vectors in the space.

The Existence and Uniqueness Theorem

Proof. Since \(\mathcal{B}\) spans \(V\), for the given \(\mathbf{v}\in V\) there are scalars \(c_1,\dots,c_n\) with \(\mathbf{v}=\sum_{i=1}^{n}c_i\mathbf{b}_i\). This proves existence. To prove uniqueness, suppose another list \(d_1,\dots,d_n\) also represents \(\mathbf{v}\). Then \(\sum_{i=1}^{n}c_i\mathbf{b}_i=\sum_{i=1}^{n}d_i\mathbf{b}_i\), so \(\sum_{i=1}^{n}(c_i-d_i)\mathbf{b}_i=\mathbf{0}\). Independence of the basis implies \(c_i-d_i=0\) for every \(i\). Hence \(c_i=d_i\) for every \(i\), and the two lists are identical. \(\square\)

The proof also identifies exactly where each basis condition enters. If the list fails to span, there is some vector with no coefficient list at all. If the list is dependent, there are coefficients \(a_i\), not all zero, for which \(\sum a_i\mathbf{b}_i=\mathbf{0}\). Any representation of a vector can then be changed by adding those coefficients: \(\mathbf{v}=\sum c_i\mathbf{b}_i=\sum(c_i+a_i)\mathbf{b}_i\). Thus dependence prevents uniqueness whenever a representation exists.

Proof. Suppose first that the list is independent. If \(\mathbf{v}\) belongs to its span, the definition of span gives at least one representation \(\mathbf{v}=\sum_{i=1}^{p}c_i\mathbf{v}_i\). If also \(\mathbf{v}=\sum_{i=1}^{p}d_i\mathbf{v}_i\), subtracting gives \(\sum_{i=1}^{p}(c_i-d_i)\mathbf{v}_i=\mathbf{0}\). Independence forces \(c_i-d_i=0\) for each \(i\), proving uniqueness. Conversely, suppose every vector in the span has exactly one representation. The zero vector is in the span, and its representation with all coefficients zero is one representation. If \(\sum_{i=1}^{p}c_i\mathbf{v}_i=\mathbf{0}\), this is another representation of the zero vector. Uniqueness forces every \(c_i=0\), so the list is independent. \(\square\)

Coordinates in \(\mathbb{R}^2\)

In \(\mathbb{R}^n\), the same procedure uses the basis vectors as columns. If \(\mathbf{A}=\begin{bmatrix}\mathbf{b}_1&\cdots&\mathbf{b}_n\end{bmatrix}\) and \(\mathbf{x}=[\mathbf{v}]_{\mathcal{B}}\), then the definition of matrix multiplication gives \(\mathbf{A}\mathbf{x}=\mathbf{v}\). The entries of \(\mathbf{x}\) are still ordered according to \(\mathcal{B}\). For other vector spaces, coefficient matching may be more natural than forming a matrix.

Coordinates of Polynomials

Coefficient matching is valid because two polynomials are equal exactly when their coefficients at each power of \(x\) agree. The order of the basis should not be silently replaced by the order of powers in the polynomial: here the first coordinate multiplies \(1\), the second multiplies \(1+x\), and the third multiplies \(1+x+x^2\).

Coordinates of Matrices

Coordinate Columns Preserve Linear Combinations

Coordinates are useful beyond describing one vector: they translate combinations of vectors into combinations of coordinate columns. This fact follows from the same uniqueness theorem, rather than being an extra assumption.

Proof. Write \(\mathbf{u}=\sum_{i=1}^{n}u_i\mathbf{b}_i\) and \(\mathbf{v}=\sum_{i=1}^{n}v_i\mathbf{b}_i\), where the coefficients are their unique coordinates. Distributivity of scalar multiplication and addition gives \(a\mathbf{u}+b\mathbf{v}=\sum_{i=1}^{n}(au_i+bv_i)\mathbf{b}_i\). By uniqueness of coordinates, the coordinate of the left-hand vector in position \(i\) is \(au_i+bv_i\). This holds for each \(i\), which is precisely the stated equality of columns. \(\square\)

Common Mistake: Changing the Basis Order

A coordinate column is not just a collection of coefficients; its entries are attached to specific basis vectors in a fixed order. In the polynomial example, the correct expression was \(5(1)-7(1+x)+4(1+x+x^2)\). A tempting but wrong calculation is to read the same column as \(-7(1)+5(1+x)+4(1+x+x^2)\), as if the first two basis vectors had been swapped. That gives \(2+9x+4x^2\), not \(2-3x+4x^2\).

The correction is to write the basis order next to the coordinate column and reconstruct the vector using matching positions: first coefficient with first basis vector, second with second, and so on. A useful check is always to perform that reconstruction. If it does not return the original vector, either the coefficients or their order is wrong.