Find a basis of the subspace W of 𝐑^4 orthogonal to u1=(1,-2,3,4) and u2=(3,-5,7,8)

Name: Problem 65, Chapter 7: Linear Algebra
Uploaded: 2021-04-21T02:30:34-08:00
Duration: 4 min 22 s
Channel: Chris Trentman
Description: Find a basis of the subspace W of 𝐑^4 orthogonal to u1=(1,-2,3,4) and u2=(3,-5,7,8)

step 1 We're given two vectors U1 in R4, the components 1 -negative 2, 3, 4, and another vector in R4,

Find a basis of the subspace W of 𝐑^4 orthogonal to...

Key Concepts

Vector Space

A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars, and which satisfy certain axioms such as associativity, distributivity, and the existence of an additive identity and inverses. The space R? is an example where each vector has four components and operations follow standard arithmetic rules involving these components.

Subspace

A subspace is a subset of a vector space that itself is a vector space under the same operations. In many problems, including this one, one finds a subset of R? that not only is closed under addition and scalar multiplication, but also satisfies additional conditions, such as being orthogonal to specified vectors.

Basis

A basis of a vector space or subspace is a minimal set of vectors that are linearly independent and span the space. Finding a basis is essential for understanding the structure of the space, as every element of the space can be uniquely represented as a linear combination of the basis vectors.

Orthogonality

Orthogonality refers to the relationship between two vectors that have a dot product of zero. This concept is central to many areas of linear algebra, including the process of finding vectors that are perpendicular to a given set of vectors.

Orthogonal Complement

The orthogonal complement of a set of vectors consists of all vectors that are orthogonal to each vector in the given set. In this context, it pertains to finding all vectors in R? that are perpendicular to the given vectors, forming a subspace whose basis is sought.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is used to define the concept of orthogonality; if the dot product of two vectors is zero, the vectors are orthogonal.

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