Question

Consider two subspaces V and Wof ${ℝ}^n$, where V is contained in W. In Exercise 62 we learned that$dim\left(V\right)\le dim\left(W\right)$ . Show that if $dim\left(V\right)=dim\left(W\right)$, then$V=W$ .

Short answer

It is proved that if $\dim \left(V\right)=\dim \left(W\right)$, then $V=W$.

Step-by-step solution

Describe the dimension of the image

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

Show that if dim(V)=dim(W), then V=W

Let$\dim \left(V\right)=\dim \left(W\right)=n$

The number of vectors in a basis of $V$is called the dimension of .$V$

There are$n$vectors in the basis of$V$and $W$.

Let$v_1,v_2,\mathrm{...},v_n$be a basis of$V$and $w_1,w_2,\mathrm{...},w_n$be a basis of$W$.

The vectors $v_1,v_2,\mathrm{...},v_n$are in $W$because $V$is contained in $W$.

If $n$vectors are linearly independent, then vectors will form a basis of $W$.

Therefore, the vectors$v_1,v_2,\mathrm{...},v_n$ form a basis of $W$.

Therefore,$V=W$ .

If $\dim \left(V\right)=\dim \left(W\right)$, then$V=W$ .

Hence, it is proved that if$\dim \left(V\right)=\dim \left(W\right)$ , then$V=W$ .