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: Recall the definition of a basis. A basis of a vector space is a linearly independent subset that spans the whole space. This is precisely what statement (C) says, so (C) is a true statement.

Step 2: Check uniqueness. A vector space (of dimension greater than zero) always has infinitely many bases. For example, in \(\mathbb{R}^2\), both \(\{(1,0),(0,1)\}\) and \(\{(1,1),(1,-1)\}\) are valid bases. So a basis is never forced to be unique, which makes statement (A) true.

Step 3: Statement (B) claims the opposite, that a basis is necessarily unique. This directly contradicts the standard result confirmed in Step 2, so (B) is false, i.e. incorrect.

Step 4: Since (A) and (C) are correct, statement (D) ("one of the statements given above is correct") is also a true statement, not the incorrect one.

Hence the only incorrect statement is (B): \(\boxed{\text{"A basis for a vector space is necessarily unique."}}\)