Suppose 𝐯1, …, 𝐯n span a subspace V ⊂ℝ^m. Prove that 𝐰 is orthogonal to V if and only if 𝐰 ∈coke

step 1 All right, so for the proof of the forward direction, we say that if our X vector is orthogonal

Suppose 𝐯1, …, 𝐯n span a subspace V ⊂ℝ^m. Prove that 𝐰 is...

Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_n$ span a subspace $V \subset \mathbb{R}^m$. Prove that $\mathbf{w}$ is orthogonal to $V$ if and only if $\mathbf{w} \in \operatorname{coker} A$, where $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ is the matrix with the indicated columns.

Key Concepts

Spanning Set and Column Space

A spanning set for a subspace is a collection of vectors such that every vector in the subspace can be expressed as a linear combination of these vectors. When the vectors are arranged as columns in a matrix, they form the column space of the matrix, which is the set of all possible linear combinations of its columns. This concept is fundamental in relating the subspace V to the matrix A in the given problem.

Kernel of the Transpose

The kernel of the transpose of a matrix, ker(A^T), consists of all vectors that yield a zero when multiplied by the transpose of the matrix. In finite-dimensional spaces with an inner product, ker(A^T) is the orthogonal complement of the column space of A, which directly establishes the equivalence between a vector being orthogonal to the subspace spanned by the columns of A and lying in the cokernel of A.

Orthogonal Complement

This concept refers to the set of all vectors in a vector space that are perpendicular to every vector in a given subspace. It is central to understanding conditions of orthogonality within inner product spaces, where a vector is said to be orthogonal to a subspace if its inner product with every vector in the subspace is zero.

Cokernel

In linear algebra, the cokernel of a matrix A (or a linear map) is defined as the quotient of the codomain by the image of A. In finite-dimensional inner product spaces, the cokernel is isomorphic to the orthogonal complement of the column space (image) of A, linking the concept to conditions of orthogonality.

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