1 −1 ) and Firstly, I have done a leftward version of the original transformation which described by the matrix ( 0 1 then I have stretched out space by 2 units in the 𝑥 direction and 3 units in the 𝑦 direction. This is described by the 2 0 ). Can you see why? (It might help if you draw it out) matrix ( 0 3 We say that the outcome of applying both of these tranformations is their composition. What you are effectively doing, is multiplying the matrices. To do this, you have to imagine they are functions (which they are) and so function notation applies (reading from right to left). For example: 𝑔(𝑓(𝑥)) This means to calculate 𝑓(𝑥) then put it into 𝑔(𝑥). The same applies for matrices: 𝑀1 𝑀2 This means do 𝑀2 first and then 𝑀1. With what I said earlier in mind, what do you think is the matrix composition of: (
2 0 1 −1 )( ) 0 3 0 1
The question is really saying “What is the overall effect of applying a rightward shear (name of the first transformation) followed by a stretching of space?” It could be put even more simply as “Record the position of 𝑖̂ and 𝑗̂ after both transformations” Let’s run through this step by step. Firstly, let’s find where 𝑖̂ lands after both transformations: 1 ( ) 𝑖𝑠 𝑤ℎ𝑒𝑟𝑒 𝑖𝑡 𝑙𝑎𝑛𝑑𝑠 𝑎𝑓𝑡𝑒𝑟 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 0 𝑤𝑒 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝑜𝑢𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑖𝑠 𝑣𝑒𝑐𝑡𝑜𝑟 𝑔𝑜𝑒𝑠 𝑎𝑓𝑡𝑒𝑟 𝑎𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑚𝑒𝑡ℎ𝑜𝑑 𝑤𝑒 ℎ𝑎𝑑 𝑒𝑎𝑟𝑙𝑖𝑒𝑟: 2 0 2 1∙( )+0∙( )= ( ) 0 3 0 𝑤𝑒 𝑡ℎ𝑒𝑛 𝑟𝑒𝑝𝑒𝑎𝑡 𝑡ℎ𝑖𝑠 𝑚𝑒𝑡ℎ𝑜𝑑 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑤ℎ𝑒𝑟𝑒 𝑗̂ 𝑙𝑎𝑛𝑑𝑠 𝑎𝑓𝑡𝑒𝑟 𝑏𝑜𝑡ℎ 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛𝑠 𝑘𝑛𝑜𝑤𝑖𝑛𝑔 1 𝑡ℎ𝑎𝑡 𝑖𝑡 𝑙𝑎𝑛𝑑𝑒𝑑 𝑎𝑡 ( ) 𝑎𝑓𝑡𝑒𝑟 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛: 1 2 0 −2 −1 ∙ ( ) + 1 ∙ ( ) = ( ) 0 3 3
𝑡ℎ𝑒𝑛, 𝑤𝑒 𝑐𝑎𝑛 𝑐𝑟𝑒𝑎𝑡𝑒 𝑎 𝑚𝑎𝑡𝑟𝑖𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑖𝑠 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑤ℎ𝑖𝑐ℎ 𝑠𝑢𝑚𝑠 𝑢𝑝 𝑏𝑜𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒 2 −2 ) 0 3
𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑚𝑎𝑡𝑖𝑜𝑛𝑠: (
If you don’t understand how we got to this point, go back to make sure you really understand how to find where a vector has moved to after a linear transformation. This is all we have done here: we have looked at where the (already transformed) 𝑖̂ and 𝑗̂ have gone to after the second transformation. Then, we have stored this information in a matrix which perfectly sums up both transformations.
21
