Tutorial Video
Addition of Scalars
Lets first look at the addition of scalars i.e. 1 by 1 matrices. Let's now for the sake of conceptualisation prescribe the values we have to objects. If we have 3 pens at home and order 5 more pens then when our order arrives we will have:
3 pens + 5 pens = 8 pens
![\displaystyle \left[ {\text{3 pens}} \right]+\left[ {\text{5 pens}} \right]=\left[ {8\text{ pens}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+%7B%5Ctext%7B3+pens%7D%7D+%5Cright%5D%2B%5Cleft%5B+%7B%5Ctext%7B5+pens%7D%7D+%5Cright%5D%3D%5Cleft%5B+%7B8%5Ctext%7B+pens%7D%7D+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \left[ 3 \right]+\left[ 5 \right]=\left[ 8 \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+3+%5Cright%5D%2B%5Cleft%5B+5+%5Cright%5D%3D%5Cleft%5B+8+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
To write this in MATLAB we would use
3+5
Addition of Column Vectors
Now let us conceptualise a more complicated example. Assume we have 3 pens and 2 pads at home, we then make a order for 5 more pens and 3 more pads.
Since pens and pads are separate objects we classify them as such. Let's essentially treat each item as a scalar:
3 pens + 5 pens = 8 pens
2 pads + 3 pads = 5 pads
![\displaystyle \left[ {\text{2 pads}} \right]+\left[ {\text{3 pads}} \right]=\left[ {5\text{ pads}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+%7B%5Ctext%7B2+pads%7D%7D+%5Cright%5D%2B%5Cleft%5B+%7B%5Ctext%7B3+pads%7D%7D+%5Cright%5D%3D%5Cleft%5B+%7B5%5Ctext%7B+pads%7D%7D+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \left[ 2 \right]+\left[ 3 \right]=\left[ 5 \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+2+%5Cright%5D%2B%5Cleft%5B+3+%5Cright%5D%3D%5Cleft%5B+5+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
If these are instead written as a column vector we will have:
![\displaystyle \left[ {\begin{array}{*{20}{c}} {3\text{ pens}} \\ {2\text{ pads}} \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} {\text{5 pens}} \\ {3\text{ pads}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {8\text{ pens}} \\ {5\text{ pads}} \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B3%5Ctext%7B+pens%7D%7D+%5C%5C+%7B2%5Ctext%7B+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D%2B%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B%5Ctext%7B5+pens%7D%7D+%5C%5C+%7B3%5Ctext%7B+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B8%5Ctext%7B+pens%7D%7D+%5C%5C+%7B5%5Ctext%7B+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \left[ {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} 5 \\ 3 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 8 \\ 5 \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+3+%5C%5C+2+%5Cend%7Barray%7D%7D+%5Cright%5D%2B%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+5+%5C%5C+3+%5Cend%7Barray%7D%7D+%5Cright%5D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+8+%5C%5C+5+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
To write this in MATLAB we would use
[3;2]+[5;3]
If I instead had 2 pens on my desk and I ordered 3 pads then in column vector notation I would have:
![\displaystyle \left[ {\begin{array}{*{20}{c}} {\text{2 pens}} \\ {0\text{ pads}} \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} {\text{0 pens}} \\ {3\text{ pads}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {2\text{ pens}} \\ {3\text{ pads}} \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B%5Ctext%7B2+pens%7D%7D+%5C%5C+%7B0%5Ctext%7B+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D%2B%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B%5Ctext%7B0+pens%7D%7D+%5C%5C+%7B3%5Ctext%7B+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B2%5Ctext%7B+pens%7D%7D+%5C%5C+%7B3%5Ctext%7B+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
With this notation bare in mind that pens and pads are separate objects meaning a pen cannot transform into a pad and a pad cannot transform into a pen!
![\displaystyle \left[ {\begin{array}{*{20}{c}} 2 \\ 0 \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} 0 \\ 3 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 2 \\ 3 \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+2+%5C%5C+0+%5Cend%7Barray%7D%7D+%5Cright%5D%2B%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+0+%5C%5C+3+%5Cend%7Barray%7D%7D+%5Cright%5D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+2+%5C%5C+3+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
To write this in MATLAB we would use
[2;0]+[0;3]
Note how we have to use zeros here to denote that we don't have a pad or pen in the first and second column vectors respectively. Element by Element operations will only work if the Arrays have the same amount of Elements.
Addition of a Matrices
Okay so far, so good. Let's make things slightly more complicated now. Assume there are two neighbours, neighbour 1 lives in the red house and he has 3 pens and 2 pads, he makes an order for 5 pens and 3 pads and neighbour 2 lives in the green house and she has 2 pens and 2 pads, she makes an order for 7 pens and 5 pads. In this case after the order the man in the red house would have 8 pens and 5 pads whilst the woman would have 9 pens and 7 pads.
![\displaystyle \left[ {\begin{array}{*{20}{c}} {\text{3 pens}} & {2\text{ pens}} \\ {\text{2 pads}} & {\text{2 pads}} \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} {\text{5 pens}} & {\text{7 pens}} \\ {\text{3 pads}} & {5\text{ pads}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\text{8 pens}} & {\text{9 pens}} \\ {5\text{ pads}} & {7\text{ pads}} \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B%5Ctext%7B3+pens%7D%7D+%26+%7B2%5Ctext%7B+pens%7D%7D+%5C%5C+%7B%5Ctext%7B2+pads%7D%7D+%26+%7B%5Ctext%7B2+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D%2B%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B%5Ctext%7B5+pens%7D%7D+%26+%7B%5Ctext%7B7+pens%7D%7D+%5C%5C+%7B%5Ctext%7B3+pads%7D%7D+%26+%7B5%5Ctext%7B+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+%7B%5Ctext%7B8+pens%7D%7D+%26+%7B%5Ctext%7B9+pens%7D%7D+%5C%5C+%7B5%5Ctext%7B+pads%7D%7D+%26+%7B7%5Ctext%7B+pads%7D%7D+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \left[ {\begin{array}{*{20}{c}} 3 & 2 \\ 2 & 2 \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} 5 & 7 \\ 3 & 5 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 8 & 9 \\ 5 & 7 \end{array}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+3+%26+2+%5C%5C+2+%26+2+%5Cend%7Barray%7D%7D+%5Cright%5D%2B%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+5+%26+7+%5C%5C+3+%26+5+%5Cend%7Barray%7D%7D+%5Cright%5D%3D%5Cleft%5B+%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D+8+%26+9+%5C%5C+5+%26+7+%5Cend%7Barray%7D%7D+%5Cright%5D&bg=ffffff&fg=000000&s=0&c=20201002)
To write this in MATLAB we would use
[3,2;2,2]+[5,7;3,5]
Now obviously we can conceive of more complicated scenarios where there are say 5 neighbours and they each have and order 7 different item types of stationary which would give a 7 by 5 matrix opposed to the 2 by 2 case above.
