A vector is always expressed in terms of *basis vectors*, which define the coordinate system. By default, we often assume an *orthogonal basis* for simplicity: $ e_{\hat i} = \begin{bmatrix} 1 \\0 \end{bmatrix} \quad e_{\hat j} = \begin{bmatrix} 0 \\1 \end{bmatrix} $ When [[Changing Basis Vectors]], we must re-express every vector $r$ relative to the new coordinate system. For orthogonal basis vectors, this can be achieved using [[Vector Projection]] or an [[Inverse Matrix]]. **Properties of Basis Vectors:** - *Linear Independence:* A vector is [[Linearly Independent Vectors|linearly independent]] if it cannot be expressed as a linear combination of the other basis vectors. - *Span:* The basis vectors must span the entire space. For $n$-dimensional space, this requires $n$ basis vectors. - *Orthogonality:* In an orthogonal basis, all basis vectors are mutually perpendicular ($b_i \cdot b_j = 0$ for $i \neq j$). Orthogonality simplifies calculations like projections and matrix inversion.