In a vector space like a Hilbert space, vectors are abstract objects. They exist independent of any coordinate system, representing concepts like force, velocity, or even quantum states. To perform calculations, however, we need a way to describe these objects using numbers. This is where a basis comes in.

Once we choose a basis, any vector in the space can be written as a unique linear combination of those basis vectors. The coefficients, or weights, of this combination form the vector's numerical representation. This list of numbers is called the coordinate vector.

The process of assigning a coordinate vector to an abstract vector is a formal mapping. For a vector space $V$ with a basis $B = \{\vec{b}_1, \vec{b}_2, \dots, \vec{b}_n\}$, we can express any vector $\vec{v} \in V$ as:

The coordinate vector of $\vec{v}$ relative to the basis $B$ is then the column vector of these weights:

This mapping is an isomorphism between the abstract vector space $V$ and the familiar coordinate space $\mathbb{R}^n$ (or $\mathbb{C}^n$ for complex spaces). It's a perfect one-to-one correspondence that preserves the vector space structure. For any given basis, this representation is unique. Change the basis, and the coordinate vector changes, but the underlying abstract vector $\vec{v}$ remains exactly the same.

The most common basis is the standard basis, which consists of vectors that point along the axes of a Cartesian coordinate system. For $\mathbb{R}^2$, the standard basis is $S = \{\vec{e}_1, \vec{e}_2\}$, where $\vec{e}_1 = (1, 0)$ and $\vec{e}_2 = (0, 1)$. The convenience here is that a vector's components are identical to its coordinates.

However, we can choose any set of linearly independent vectors as our basis. Let's say we pick a new basis, $B = \{\vec{b}_1, \vec{b}_2\}$. To find the coordinates of our vector $\vec{v}$ in this new basis, we must find the unique scalars $c_1$ and $c_2$ that satisfy $\vec{v} = c_1\vec{b}_1 + c_2\vec{b}_2$. As the graphic above shows, the coordinate vector changes to reflect the new basis, even though the geometric vector $\vec{v}$ is stationary.

This ability to change basis is not just an academic exercise. It is a powerful tool in linear algebra. Many problems become much simpler to solve by switching to a basis that is better suited to the problem's geometry or structure, such as an eigenbasis for analyzing linear transformations.

Understanding coordinate vectors bridges the gap between the abstract theory of vector spaces and the concrete calculations of matrix algebra. A vector is a geometric entity; its coordinates are just its shadow projected onto a chosen set of axes.

This fundamental link allows us to use the tools of matrix multiplication to perform complex transformations and change-of-basis calculations, translating abstract vector operations into tangible arithmetic.