Which statement about coordinate vectors in a basis is false?

Key concept: Coordinate vectors are one of the cleanest ideas in linear algebra: once a basis is fixed, every vector has exactly one coordinate representation. That uniqueness is what drives the answer here.

Step by step

Let $B=\{\vec b_1,\vec b_2,\vec b_3,\vec b_4\}$ be a basis for $\mathbb R^4$.

For any vector $\vec v\in\mathbb R^4$, the coordinate vector $C_B(\vec v)$ is the unique vector of scalars such that

$\vec v=c_1\vec b_1+c_2\vec b_2+c_3\vec b_3+c_4\vec b_4.$

So:

• A is true: $C_B(\vec v)$ is unique.
• C is true: $C_B(\vec v)\in\mathbb R^4$.
• D is true: $C_B(\vec v)=\vec 0$ iff $\vec v=\vec 0$.
• B is false: the coordinate vector is not non-unique.

Therefore the false statement is

$\boxed{B}.$

Pitfall alert

Students sometimes mix up the vector $\vec v$ with its coordinate vector $C_B(\vec v)$. The coordinate vector lives in $\mathbb R^4$ as a list of coefficients, and because $B$ is a basis, that list is unique.

Try different conditions

If $B$ were only a spanning set and not a basis, uniqueness could fail. In that case, a vector might have more than one coordinate-like representation. The basis condition is what guarantees a single answer.

Further reading

coordinate vector, basis, linear independence