Let W ⊂V with dim V=n. Suppose 𝐰1, …, 𝐰m is an orthogonal basis for W and 𝐰m+1, …, 𝐰n is an orth

step 1 In this question we have S is equals to V1, V2 and so on VN and it is given that W is orthogonal

Let W ⊂V with dim V=n. Suppose 𝐰1, …, 𝐰m is an orthogonal...

Let $W \subset V$ with $\operatorname{dim} V=n$. Suppose $\mathbf{w}_1, \ldots, \mathbf{w}_m$ is an orthogonal basis for $W$ and $\mathbf{w}_{m+1}, \ldots, \mathbf{w}_n$ is an orthogonal basis for $W^{\perp}$. (a) Prove that the combination $\mathbf{w}_1, \ldots, \mathbf{w}_n$ forms an orthogonal basis of $V$. (b) Show that if $\mathbf{v}=c_1 \mathbf{w}_1+\cdots+c_n \mathbf{w}_n$ is any vector in $V$, then its orthogonal decomposition $\mathbf{v}=\mathbf{w}+\mathbf{z}$ is given by $\mathbf{w}=c_1 \mathbf{w}_1+\cdots+c_m \mathbf{w}_m \in W$ and $\mathbf{z}=c_{m+1} \mathbf{w}_{m+1}+\cdots+c_n \mathbf{w}_n \in W^{\perp}$.

Key Concepts

Orthogonal Projection

Orthogonal projection is the process of decomposing a vector into two parts: one that lies in a given subspace and another that lies in the orthogonal complement of that subspace. This process is central in many applications such as solving least squares problems, as it provides the closest approximation of a vector within a subspace by minimizing the error in the projection.

Direct Sum Decomposition

Direct sum decomposition refers to the representation of a vector space as a sum of two subspaces such that each vector in the space can be uniquely written as the sum of one vector from each subspace. When dealing with a subspace and its orthogonal complement, this decomposition allows for a clear and unique split of any vector into components that lie within the subspace and its complement.

Orthogonal Basis

An orthogonal basis of a vector space is a set of vectors that are mutually perpendicular with respect to the inner product defined on the space and are linearly independent. This property greatly simplifies the process of representing any vector in the space as a sum of basis vectors, particularly because the coefficients in the expansion can be easily computed using inner products.

Orthogonal Complement

The orthogonal complement of a subspace W in a vector space V, denoted by W?, is the set of all vectors in V that are orthogonal to every vector in W. This concept is essential as it provides a way to partition the vector space into two mutually orthogonal parts, which is crucial in various mathematical and applied contexts.

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