Prove following theorem: Let V be a vector space. If W V then W is a subspace of

Question

Prove following theorem: Let V be a vector space. If W V then W is a subspace of V if and only if the following conditions hold. (i) 0 W where 0 is the neutral element of V . (ii) If u, v W then u + v W. (iii) If u W and k R then ku W Prove following theorem: Let V be a vector space. If W V then W is a subspace of V if and only if the following conditions hold. (i) 0 W where 0 is the neutral element of V . (ii) If u, v W then u + v W. (iii) If u W and k R then ku W (i) 0 W where 0 is the neutral element of V . (ii) If u, v W then u + v W. (iii) If u W and k R then ku W

Explanation / Answer

We are given that W V. Thus, W is a subspace of V if and only if W is itself a vector space.

As per the definition of a vector space, the conditions that a vector space V must satisfy are as under:

1. For all X, Y V, X+YV( closure under vector addition).

2. For all X, Y , X+Y = Y+X ( commutativity of vector addition).     

3. For all X, Y, Z , (X+Y)+Z=X+(Y+Z) (Associativity of vector addition).

4. For all x, 0+X = X+0 = X ( Existence of Additive identity)

5. For any X, there exists a -X such that X+(-X)= 0 (Existence of additive inverse)

6. For any scalar k and the vector v V, the vector kv V( closure under scalar multiplication).

7. For all scalars r and vectors X,Y, r(X+Y)=rX+rY (Distributivity of vector addition).

8. For all scalars r,s and vectors X , (r+s)X=rX+sX (Distributivity of scalar addition).

9. For all scalars r,s and vectors X, r(sX)=(rs)X( Associativity of scalar multiplication).

10. For all vectors X, 1X=X ( Existence of Scalar multiplication identity).

Now, it may be observed that if W satisfies the given 3 conditions, then W also satiesfies all the other axioms of a vector space so that W is itself a vector space.

Conversely, if W is a vector space, then , by virtue of the axioms of a vector space, W satisfies the given 3 conditions.

Hence, if W V then W is a subspace of V if and only if the following conditions hold.

(i) 0 W where 0 is the neutral element of V .

(ii) If u, v W then u + v W.

(iii) If u W and k R then ku W

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.