Question

Consider a subspace V of${ℝ}^n$with $dim\left(V\right)=n$. Explain why $V={ℝ}^n$.

Short answer

$V={ℝ}^n$because $V$ is subspace of${ℝ}^n$ .

Step-by-step solution

Describe the dimension of the image

For any matrix A,

$dim\left(\text{im\hspace{0.17em}}A\right)=\text{rank}\left(A\right)$

Explain the reason why V=ℝn

Consider that$V$ in${ℝ}^n$ and $\dim \left(V\right)=n$.

This means that there are exactly$n$ linearly independent vectors${\overrightarrow{v}}_1,\mathrm{...},{\overrightarrow{v}}_n$ that form a basis of $V$.

From the previous exercise, it is known that if$\dim V=\dim W$ and$V$ is subspace of$W$ then$V=W$ .

Here, it is known that$V$ is subspace of ${ℝ}^n$and that $\dim V=\dim {ℝ}^n$, therefore, $V={ℝ}^n$.

Hence,$V={ℝ}^n$because $V$is subspace of${ℝ}^n$ .