Question

If V and W are subspaces of $R^n$then their union role="math" localid="1664184474463" $VUW$must be a subspace of $R^n$as well.

Short answer

The above statement is false.

If V and W are subspaces of $R^n$then their union$V\cup W$ may be a subspace of $R^n$iff one subspace is contained in the other.

Step-by-step solution

One-step test of a subspace

Let be a vector space over the field K, and let V is any subset of$R^n$ , then V is a subspace of $R^n$iff-

1. $0\in V$
2. $\overrightarrow{u},\overrightarrow{v}\in V,\text{a,b}\in F$
   

$\Rightarrow a\overrightarrow{u}+b\overrightarrow{v}\in V$

Considering an example

Let us consider two sets$V=\left\{\left(a,0\right):a\in \mathrm{ℝ}\right\}\text{and W}=\left\{\left(0,b\right):b\in \mathrm{ℝ}\right\}.$

Let$R^2$ be a vector space over the field

Since $V\text{and}W$satisfies all the conditions given in step (1), therefore,$V\text{and}W$ are two subspaces of$R^2$

Finding the union of V and W

Let$\overrightarrow{u}=\left(1,0\right)\in Vand\overrightarrow{v}=\left(0,1\right)\in W.$

Then u and v both belong to the union$V\cup W.$

But$\overrightarrow{u}+\overrightarrow{v}=(1,0)+(0,1)=(1,1)\notin V\cup W.$

Hence,$V\cup W$ is not a subspace of$R^2$

Final Answer

Since we found an example for n = 2 which shows that the union of two subspaces $V\text{and}W$is not a subspace of vector space $R^2$, therefore it will hold for every value of ‘n’ and hence the statement “If V and W are subspaces of $R^n$then their union $V\cup W$may be a subspace of $R^n$iff one subspace is contained in the other” is false.