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Vector Space : Properties of Vector Space - Linear Algebra Video Lecture | Engineering Mathematics - Engineering Mathematics
FAQs on Vector Space : Properties of Vector Space - Linear Algebra Video Lecture - Engineering Mathematics - Engineering Mathematics
| 1. What are the main properties of a vector space? |  |
Ans. The main properties of a vector space include closure under addition and scalar multiplication, the existence of a zero vector, the existence of additive inverses, and the satisfaction of several axioms such as associativity and commutativity of vector addition, distributive properties, and the identity element for scalar multiplication.
| 2. How do you determine if a set is a vector space? | |
Ans. To determine if a set is a vector space, you must check if it satisfies the vector space axioms. This includes verifying closure under addition and scalar multiplication, the existence of a zero vector, and that every vector has an additive inverse. If all these conditions are met, the set forms a vector space over a specified field.
| 3. Can you provide an example of a vector space? | |
Ans. An example of a vector space is the set of all ordered pairs of real numbers, denoted as \( \mathbb{R}^2 \). In this space, two vectors can be added together and multiplied by a scalar, and all vector space properties hold true, such as commutativity, associativity, and the existence of a zero vector (the pair (0,0)).
| 4. What is the significance of the zero vector in a vector space? | |
Ans. The zero vector is significant in a vector space because it serves as the identity element for vector addition. This means that adding the zero vector to any vector in the space leaves the vector unchanged. Additionally, every vector must have an additive inverse, which pairs with the zero vector to satisfy the properties of a vector space.
| 5. How do the concepts of linear independence and span relate to vector spaces? | |
Ans. Linear independence refers to a set of vectors in a vector space that do not express any vector as a linear combination of the others. The span of a set of vectors is the set of all possible linear combinations of those vectors. Together, these concepts help define the dimensionality of a vector space, as the dimension is the number of vectors in a maximal linearly independent set, which spans the entire space.
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