Question

Consider two subspaces V and Wof${ℝ}^n$ , with .role="math" localid="1660906447344" $V\cap W=\left\{\overrightarrow{0}\right\}$
What is the relationship among $dim\left(V\right)$, $\dim \left(W\right)$, and$dim(V+W)$ ? (For the definition of $V+W$, see Exercise 3.2.50; Exercise 3.2.51 is helpful.) 1

Short answer

The required relationship is$\dim (V+W)=k+l=\dim V+\dim W$ .

Step-by-step solution

Describe the dimension of the image

For any matrix A,

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

Find the relationship among dim(V), dim(W), and dim(V+W)

It is known that is $B_W=\{w_1,w_2,\mathrm{...},w_k\}$base of$W$ and$B_V=\{v_1,v_2,\mathrm{...},v_l\}$ is base of $V$then vector form $B_W\cup B_V=\{w_1,w_2,\mathrm{...},w_k,v_1,v_2,\mathrm{...},v_l\}$are linearly independent.

Now, it is known that since they are independent then they form a basis from$k+l$ dimensional space, and it is known that they span $W+V$. Therefore,

$\dim (V+W)=k+l=\dim V+\dim W$

Hence, the required relationship is .$\dim (V+W)=k+l=\dim V+\dim W$