Question

Which of the following statements is incorrect?

• A. A basis for a vector space need not be unique.
• B. A basis for a vector space is necessarily unique.
• C. If each vector of a vector space is a linear combination of elements of a subset of the vector space where the subset is linearly independent, then the subset is called a basis set of the vector space.
• D. One of the statements given above is correct.

Hint:

Recall that a vector space of positive dimension always has infinitely many different bases, so uniqueness of a basis is false.

Answer 1

The Correct Option is B

Step 1: Test each option against a concrete example. Take $V = \mathbb{R}^2$.

Step 2: The set $\{(1,0),(0,1)\}$ is a basis of $\mathbb{R}^2$ since it is linearly independent and spans $V$. This matches the description in option (C), so (C) holds.

Step 3: The set $\{(1,1),(1,-1)\}$ is also linearly independent (neither is a scalar multiple of the other) and also spans $\mathbb{R}^2$, since any $(x,y)$ can be written as a combination of these two vectors. So $\mathbb{R}^2$ already has at least two different bases, proving option (A) is true and directly contradicting option (B).

Step 4: Since option (B) says a basis "is necessarily unique," and we just exhibited two different bases of the same space, option (B) is false.

Step 5: Because (A) and (C) are true, option (D) is also a correct statement, so it cannot be the incorrect one asked for.

\[\boxed{\text{Option (B) is the incorrect statement}}\]