Vector Space : Properties of Vector Space - Linear Algebra Video Lecture | Engineering Mathematics - Civil Engineering (CE)

Ans. The properties of a vector space are as follows: 1. Closure under addition: For any two vectors u and v in a vector space, their sum u + v is also in the vector space. 2. Closure under scalar multiplication: For any scalar c and any vector v in a vector space, the product cv is also in the vector space. 3. Associativity of addition: For any three vectors u, v, and w in a vector space, (u + v) + w = u + (v + w). 4. Commutativity of addition: For any two vectors u and v in a vector space, u + v = v + u. 5. Identity element of addition: There exists a vector 0 such that for any vector v in a vector space, v + 0 = v. 6. Inverse element of addition: For any vector v in a vector space, there exists a vector -v such that v + (-v) = 0. 7. Distributivity of scalar multiplication over vector addition: For any scalar c and any vectors u and v in a vector space, c(u + v) = cu + cv. 8. Distributivity of scalar multiplication over scalar addition: For any scalars c and d and any vector v in a vector space, (c + d)v = cv + dv. 9. Associativity of scalar multiplication: For any scalars c and d and any vector v in a vector space, (cd)v = c(dv). 10. Identity element of scalar multiplication: For any vector v in a vector space, 1v = v.

Ans. To prove closure under addition in a vector space, we need to show that for any two vectors u and v in the vector space, their sum u + v is also in the vector space. This can be done by following these steps: 1. Assume u and v are vectors in the vector space. 2. Show that u + v satisfies all the properties of a vector space, including closure under addition. 3. Use the properties of the vector space, such as associativity and commutativity of addition, to simplify the expression u + v. 4. Show that the simplified expression is also a vector in the vector space, thereby proving closure under addition.

Ans. The identity element of addition, denoted as 0, is a special vector in a vector space. It plays a significant role in vector operations. The significance of the identity element of addition can be understood as follows: 1. It acts as a neutral element: When added to any vector v in the vector space, the identity element of addition does not change the value of v. In other words, v + 0 = v. This property is similar to the role of zero in arithmetic operations. 2. It ensures the existence of additive inverses: The identity element of addition allows for the existence of additive inverses in a vector space. For any vector v, there exists a vector -v such that v + (-v) = 0. This property ensures that vectors can be "cancelled out" by adding their additive inverses, similar to subtracting a number from itself in arithmetic. 3. It helps define vector subtraction: Vector subtraction can be defined as the addition of a vector and its additive inverse. The identity element of addition provides a reference point for defining and understanding vector subtraction.

Ans. The distributive properties of scalar multiplication in a vector space are as follows: 1. Distributivity of scalar multiplication over vector addition: For any scalar c and any vectors u and v in a vector space, c(u + v) = cu + cv. This property states that when a scalar is multiplied by the sum of two vectors, it is equivalent to multiplying the scalar with each vector separately and then adding the results. 2. Distributivity of scalar multiplication over scalar addition: For any scalars c and d and any vector v in a vector space, (c + d)v = cv + dv. This property states that when a vector is multiplied by the sum of two scalars, it is equivalent to multiplying the vector by each scalar separately and then adding the results. These distributive properties are fundamental in performing calculations and simplifications involving scalar multiplication in a vector space.

Ans. To prove the existence of an inverse element of addition in a vector space, we need to show that for any vector v in the vector space, there exists a vector -v such that v + (-v) = 0. This can be done by following these steps: 1. Assume v is a vector in the vector space. 2. Show that -v satisfies the properties of an additive inverse. This means that v + (-v) = 0. 3. Use the properties of the vector space, such as associativity and commutativity of addition, to simplify the expression v + (-v). 4. Show that the simplified expression is equal to the identity element of addition, denoted as 0, thereby proving the existence of an inverse element of addition. By proving the existence of an inverse element of addition, we establish that every vector in the vector space has a corresponding additive inverse, allowing for the cancellation of vectors through addition.