Let v be a vector over a field

Mart Hoert

Mart Hoert

Answered question


Let v be a vector over a field F with zero vector  0 and let s,T be a substance of V .then which of the following statements are false

Answer & Explanation

Don Sumner

Don Sumner

Skilled2023-05-23Added 184 answers

To evaluate the truth value of the statements, we need to consider the properties and definitions of vectors and subsets in a vector space. Let's examine each statement individually:
Statement 1: If 0āˆˆS, then S is a subspace of V.
This statement is true. For S to be a subspace of V, it must satisfy three conditions:
1. 0āˆˆS (the zero vector is an element of S),
2. S is closed under vector addition (if š® and šÆ are in S, then š®+šÆ is also in S), and
3. S is closed under scalar multiplication (if š® is in S and c is a scalar, then cš® is also in S).
Statement 2: If S is a subspace of V, then 0āˆˆS.
This statement is true. As mentioned in the explanation of Statement 1, for S to be a subspace of V, it must contain the zero vector 0.
Statement 3: If SāˆŖT={0}, then S and T are subspaces of V.
This statement is false. To demonstrate this, we can provide a counterexample. Consider V as the vector space ā„2 and let S={(1,0)} and T={(0,1)}. Both S and T are subsets of V, and their union is {(1,0),(0,1)}, which does not equal {0}. However, S and T are not subspaces of V because they do not contain the zero vector 0.
To summarize:
- Statement 1 is true.
- Statement 2 is true.
- Statement 3 is false.

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