91影视

Since we know that

Dim (Im(A)) = rank(A) and dim (Im(B)) = rank(B)

So, dim (Im(A+B)) = rank(A+B)

Let$A=\left[{\stackrel{鈫�}{v}}_1{\stackrel{鈫�}{v}}_2...{\stackrel{鈫�}{v}}_m\right]\mathrm{and}B=[w{鈨�}_{1}w{鈨�}_2...w{鈨�}_m]$

Where$v{鈨�}_{i}andw{鈨�}_i$ are column vectors of A and B respectively.

Then,

role="math" localid="1659355348377" $$\begin{gathered} \mathrm{rank}\left(\mathrm{A}\right)=\dim \left(\mathrm{span}\right(\mathrm{v}{鈨�}_1,\mathrm{v}{鈨�}_2,...,\mathrm{v}{鈨�}_{\mathrm{m}}\left)\right) \\ \mathrm{and} \\ \mathrm{rank}\left(\mathrm{B}\right)=\dim \left(\mathrm{span}\right(\mathrm{w}{鈨�}_1,\mathrm{w}{鈨�}_2,...\mathrm{w}{鈨�}_{\mathrm{m}}\left)\right) \end{gathered}$$

$鈬�rank(A+B)=dim\left(span\right(v{鈨�}_1+w{鈨�}_1,v{鈨�}_2+w{鈨�}_2,...v{鈨�}_m+w{鈨�}_m\left)\right)$

Let us claim,

$span(v{鈨�}_1+w{鈨�}_1,\stackrel{鈫�}{v}{鈨�}_2+w{鈨�}_2,...,v{鈨�}_m+w{鈨�}_m)鈯�span(v{鈨�}_1,v{鈨�}_2,...,v{鈨�}_m)+span(w{鈨�}_1,w{鈨�}_2,...,w{鈨�}_m)$

Then any vector can be written as

$$\begin{gathered} x鈨�=a_1(v{鈨�}_1+w{鈨�}_1)+a_2(v{鈨�}_2+w{鈨�}_2)+...+a_m(v{鈨�}_m+w{鈨�}_m) \\ 鈬�x鈨�=(a_{1}v{鈨�}_1+a_{2}v{鈨�}_2+...+a_{m}v{鈨�}_m)+(a_{1}w{鈨�}_1+a_{2}w{鈨�}_2+...+a_{m}w{鈨�}_m) \\ 鈬�x鈨�鈭�span(v{鈨�}_1,v{鈨�}_2,...,v{鈨�}_m)+span(w{鈨�}_1,w{鈨�}_2,...,w{鈨�}_m) \end{gathered}$$

for some scalars $a{鈨�}_1,a{鈨�}_2,...,a_m$.

$$\begin{gathered} 鈬�span(v{鈨�}_1+w{鈨�}_1,v{鈨�}_2+w{鈨�}_2,...,v{鈨�}_m+w{鈨�}_m)鈯�span(v{鈨�}_1,v{鈨�}_2,...,v{鈨�}_m)+span(w{鈨�}_1,w{鈨�}_2,...,w{鈨�}_m) \\ 鈬�Im(A+B)鈯�Im\left(A\right)+Im\left(B\right) \\ 鈬�dim\left(Im\right(A+B\left)\right)鈮�dim\left(Im\right(A\left)\right)+dim\left(Im\right(B\left)\right) \\ 鈬�rank(A+B)鈮�rank\left(A\right)+rank\left(B\right) \end{gathered}$$

If we have two n x m matrices A and B then

$$\begin{gathered} 鈬�Im(A+B)鈯�Im\left(A\right)+Im\left(B\right) \\ 鈬�dim\left(Im\right(A+B\left)\right)鈮�dim\left(Im\right(A\left)\right)+dim\left(Im\right(B\left)\right) \\ 鈬�rank(A+B)鈮�rank\left(A\right)+rank\left(B\right) \end{gathered}$$