Let W ⊂V be a subspace. A subspace Z ⊂V is called a complementary subspace to W if (i) W ∩Z={0},

Step 1: **Show that the $x$u002D and $y$u002Daxes are complementary subspaces of $u005Cmathbb{R}^2$.**u000Au002D Def

Let W ⊂V be a subspace. A subspace Z ⊂V is called a...

Let $W \subset V$ be a subspace. A subspace $Z \subset V$ is called a complementary subspace to $W$ if (i) $W \cap Z=\{0\}$, and (ii) $W+Z=V$, i.e., every $\mathbf{v} \in V$ can be written as $\mathbf{v}=\mathbf{w}+\mathbf{z}$ for $\mathbf{w} \in W$ and $\mathbf{z} \in Z$. (a) Show that the $x$ - and $y$-axes are complementary subspaces of $\mathbb{R}^2$. (b) Show that the lines $x=y$ and $x=3 y$ are complementary subspaces of $\mathbb{R}^2$. (c) Show that the line $(a, 2 a, 3 a)^T$ and the plane $x+2 y+3 z=0$ are complementary subspaces of $\mathbb{R}^3$. (d) Prove that if $\mathbf{v}=\mathbf{w}+\mathbf{z}$, then $\mathbf{w} \in W$ and $\mathbf{z} \in Z$ are uniquely determined.

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Key Concepts

Subspace

A subspace is a subset of a vector space that is itself a vector space under the operations of addition and scalar multiplication. It must include the zero vector and be closed under these operations, forming an essential building block in linear algebra.

Complementary Subspaces

Complementary subspaces are two subspaces whose sum gives the entire vector space and whose intersection is only the zero vector. This ensures that every vector in the space can be decomposed as the sum of one vector from each subspace, thus partitioning the space in a meaningful way.

Direct Sum Decomposition

Direct sum decomposition is a method of expressing a vector space as the sum of two or more subspaces such that each element of the vector space is uniquely represented as a combination of elements from these subspaces. When subspaces are complementary, their direct sum recovers the original vector space.

Uniqueness of Decomposition

The uniqueness of a vector decomposition in the context of complementary subspaces is guaranteed by the condition that their intersection is trivial. This means that if a vector is expressed as the sum of two components from the subspaces, each component is uniquely determined, a property that underpins many linear algebraic constructions.

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