Which sets keep every linear combination?

Source scope: §3.1.2. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Why the origin matters

The line $S=\{(t,2t):t\in\mathbb R\}$ contains zero. Adding two points adds their parameters, and scaling a point scales its parameter. Neither operation leaves the line. The shifted line $y=2x+1$ fails immediately: it lacks zero.

A subset $S$ of a vector space $V$ is a subspace if $0\in S$, $x+y\in S$ whenever $x,y\in S$, and $\alpha x\in S$ whenever $x\in S$, $\alpha\in\mathbb R$. Both $V$ and the trivial space $\{0\}$ are subspaces.

Sum versus direct sum

For subspaces $U,W\subseteq V$,

This is a subspace: zero is $0+0$; addition regroups the two $U$ terms and the two $W$ terms; scaling distributes into both. If $U\cap W=\{0\}$, the sum is a direct sum, written $U\oplus W$.

Every vector in a direct sum has a unique decomposition. To see why, if $u+w=u'+w'$, then $u-u'=w'-w$ lies in both spaces. A zero intersection forces both differences to vanish. Conversely, a nonzero $z\in U\cap W$ gives two decompositions $z=z+0=0+z$, so uniqueness fails. This argument is a teaching expansion of the source statement.

The two coordinate axes give a direct sum of the plane. Two copies of the same axis do not: $(2,0)$ can be split as $(1,0)+(1,0)$ or $(2,0)+(0,0)$.

Count dimensions without double counting

For finite-dimensional subspaces,

The source refers to [4], Sheldon Axler, Linear Algebra Done Right, third edition, Springer International Publishing, 2015, for proof. It does not provide that proof here. In a direct sum the intersection has dimension zero, giving $\dim(U\oplus W)=\dim U+\dim W$.

Two distinct planes through zero in $\mathbb R^3$ meet in a line: $2+2-1=3$. Their sum is all of space, but their decomposition is not unique.

Explore in three dimensions

Transfer

Is the first quadrant a subspace?