1. The problem is to express a vector or a linear transformation in canonical form, which typically means representing it in a standard or simplest form. 2. In linear algebra, the canonical form often refers to the diagonal form or Jordan normal form of a matrix, or expressing a vector in terms of a canonical basis. 3. For a vector space, the canonical basis is a set of vectors where each vector has a 1 in one coordinate and 0 in all others. 4. To express a vector $\mathbf{v}$ in canonical form, write it as a linear combination of the canonical basis vectors: $$\mathbf{v} = v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2 + \cdots + v_n \mathbf{e}_n$$ where $v_i$ are the components of $\mathbf{v}$. 5. For a matrix, the canonical form depends on the context: for diagonalizable matrices, find eigenvalues and eigenvectors to diagonalize it; for others, use Jordan form. 6. The key rules are: the canonical form simplifies the structure, making properties like eigenvalues explicit. 7. Without a specific vector or matrix given, the general approach is to identify the canonical basis and express the object accordingly. Final answer: The canonical form of a vector is its coordinate representation in the canonical basis, and for matrices, it is the simplest form revealing its structure, such as diagonal or Jordan form.