Show that the space V={(x1, x2, x3) ∈𝔽^3 |x1+2 x2+2 x3=0} forms a vector space.

Name: Problem 2, Chapter 4: Linear Algebra
Uploaded: 2020-10-22T13:06:52-08:00
Duration: 1 min 37 s
Channel: Dharmendra Jain
Description: Show that the space V={(x1, x2, x3) ∈𝔽^3 |x1+2 x2+2 x3=0} forms a vector space.

step 1 Since V1 vector V2 vector and V3 vector are mutually orthogonal non -zero vectors, so we can wri

Show that the space V={(x1, x2, x3) ∈𝔽^3 |x1+2 x2+2 x3=0}...

Key Concepts

Vector Space

A vector space is a collection of objects, called vectors, along with two operations (vector addition and scalar multiplication) that follow specific axioms such as associativity, commutativity, existence of an additive identity, and distributivity of scalar multiplication. This general framework is fundamental in linear algebra and provides the setting for studying linear combinations, spans, and dimensionality.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same operations. To verify that a subset is a subspace, it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. Demonstrating that a set defined by a linear equation satisfies these properties shows that it is a vector space in its own right.

Closure Properties

Closure under addition and scalar multiplication are essential criteria for a set to be a subspace. Specifically, if two vectors in the set add together to yield another vector in the set, and if any scalar multiple of a vector in the set is also in the set, then these operations preserve the structure of the vector space. These properties are key when showing that the solution set of a linear equation is a subspace.

Linear Homogeneous Equation

A linear homogeneous equation, such as one in which the sum of scalar multiples of variables equals zero, defines a set that includes the zero vector and tends to be closed under addition and scalar multiplication. This connection is important because the set of solutions to a homogeneous linear equation naturally forms a subspace, as all solutions satisfy the property needed for a vector space.

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