Definition: A subset \(W\) of a vector space \(V\) is a subspace if \(W\) is itself a vector space under the same operations of addition and scalar multiplication defined on \(V\).
Subspace test: A nonempty subset \(W \subseteq V\) is a subspace if and only if:
- Closed under addition: If \(\mathbf{u}, \mathbf{v} \in W\), then \(\mathbf{u} + \mathbf{v} \in W\)
- Closed under scalar multiplication: If \(\mathbf{v} \in W\) and \(c\) is a scalar, then \(c\mathbf{v} \in W\)
(These two conditions guarantee \(\mathbf{0} \in W\) and that additive inverses exist.)
Examples:
- \(\{\mathbf{0}\}\) is a subspace of every vector space (the trivial subspace)
- Every vector space \(V\) is a subspace of itself
- In \(\mathbb{R}^3\), any line or plane through the origin is a subspace
- The set \(\{(x,y,0) : x,y \in \mathbb{R}\}\) is a subspace of \(\mathbb{R}^3\)
Non-example: The set \(\{(x,y,1) : x,y \in \mathbb{R}\}\) is NOT a subspace of \(\mathbb{R}^3\) because it does not contain the zero vector.
Children
- Definition of subspaceA subset $W$ of a vector space $V$ is a subspace if it contains $0$ and is closed under addition and scalar...
- Definition of subspace spanned by a set of a set of vectorsThe subspace spanned by $S$ is $\operatorname{span}S$, all finite linear combinations of its elements. It contains...
- Definition of the 0/trivial subspaceThe zero, or trivial, subspace is $\{0\}$. It is closed under every linear combination and has dimension zero. Its...
- Definition of 0/trivial subspaceThe trivial subspace of any vector space is $\{0 V\}$. It has dimension zero and the empty basis. It is contained in...
- A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinationsA nonempty subset $W\subseteq V$ is a subspace iff $au+bv\in W$ for all $u,v\in W$ and scalars $a,b$. Taking $a=b=0$...
- Definition of intersection of subspacesThe intersection $\bigcap iW i$ consists of vectors belonging to every listed subspace. It is a subspace. For...
- The intersection of subspaces is a subspaceEvery subspace contains zero, so their intersection does too. If $x,y$ lie in every $W i$, each combination $ax+by$...
- Definition of sum of subspacesThe sum $U+W=\{u+w:u\in U,w\in W\}$ is the smallest subspace containing $U\cup W$. More generally, a sum of a family...
- The sum of subspaces is a subspaceIf $x=u 1+w 1$ and $y=u 2+w 2$ belong to $U+W$, then $ax+by=(au 1+bu 2)+(aw 1+bw 2)$ also belongs. Zero is included....
- Definition of direct sum of subspacesThe sum $U+W$ is direct, written $U\oplus W$, if every vector in the sum has a unique decomposition $u+w$. For two...
- The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.For finite-dimensional subspaces with direct sum, $\dim(U 1\oplus\cdots\oplus U k)=\sum i\dim U i$. Combine bases of...
- Definition of independent subspacesSubspaces $U 1,\ldots,U k$ are independent if $u 1+\cdots+u k=0$, with $u i\in U i$, forces each $u i=0$. This is the...
- A vector can be written uniquely as a linear combination of vectors from independent subspaces.For independent subspaces, each vector in their sum has a unique decomposition $v=u 1+\cdots+u k$ with $u i\in U i$....
- The union of bases from independent subspaces is a basis for the space.If $V=U 1\oplus\cdots\oplus U k$, combining a basis of each $U i$ yields a basis of $V$. Spanning follows from the...
- Definition of complement of a subspaceA complement of $U\subseteq V$ is a subspace $W$ with $V=U\oplus W$. In finite dimensions, extend a basis of $U$ to a...
- Theorem characterizing when a space is the direct sum of two subspacesFor subspaces $U,W\subseteq V$, $V=U\oplus W$ iff $U+W=V$ and $U\cap W=\{0\}$. The first condition gives existence of...
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