Determine a basis for the subspace W ⊂ℝ^4 consisting of all vectors which are orthogonal to the

step 1 So here in this question, the objective is to determine the basis for the subspace and that has

Determine a basis for the subspace W ⊂ℝ^4 consisting of all...

Key Concepts

Subspace

A subspace is a subset of a vector space that is also a vector space under the same operations of addition and scalar multiplication. In the context of the problem, the set of all vectors that are orthogonal to a given vector forms a subspace of ?? because it is closed under addition and scalar multiplication.

Orthogonality

Orthogonality in a vector space refers to the concept of two vectors having a dot product of zero, meaning they are perpendicular to each other. The subspace in the problem consists of vectors that are orthogonal to a specific vector, which means each vector in the subspace has a zero dot product with that vector.

Dot Product

The dot product is a scalar product of two vectors that, in ??, is computed as the sum of the products of their corresponding components. It is used to define orthogonality; if the dot product is zero, the vectors are perpendicular. This operation is fundamental in forming the constraint that defines the subspace in the problem.

Basis

A basis of a vector space or subspace is a set of linearly independent vectors that span the entire space. For the subspace defined by the orthogonality condition, a basis is a minimal set of vectors such that any vector in the subspace can be expressed as a linear combination of these basis vectors.

Dimension

The dimension of a vector space or subspace is the number of vectors in a basis for that space. In the problem, since the subspace of vectors orthogonal to a single nonzero vector in ?? is defined by one linear equation, its dimension is 3, indicating that any basis for the subspace will consist of three linearly independent vectors.

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