# One more linear transformation

#### war485

1. The problem statement, all variables and given/known data

M22 ---> R is a linear transformation.

given:

T[ 1 0 ] = 1
,,[ 0 0 ]

T[ 1 1 ] = 2
,,[ 0 0 ]

T[ 1 1 ] = 3
,,[ 1 0 ]

T[ 1 1 ] = 4
,,[ 1 1 ]

find
T[ 1 3 ]
,,[ 4 2 ]

and
T[ a b ]
,,[ c d ]

2. Relevant equations

none.

3. The attempt at a solution

I don't know if it is valid to "add" the transformations. I'm tempted to add the given linear transformations like 2T + 2T - 1T - 2T which gives
T[ 1 3 ]
,,[ 4 2 ]
which I believe it is 10. Is that valid?

T[ a b ]
,,[ c d ]
Now this one is really making me stuck. It looks like something recursive but I can't wrap my head around it. Is there something special to this?

#### wywong

T[ 1 0 ] = 1
,,[ 0 0 ]

T[ 1 1 ] = 2
,,[ 0 0 ]

T[ 1 1 ] = 3
,,[ 1 0 ]

T[ 1 1 ] = 4
,,[ 1 1 ]
If you subtract the third equation from the fourth, you get
$$T$\left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right)=1$$$

Similarly you get

$$T$\left( \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right) = T \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right) = 1$$$

I don't know if it is valid to "add" the transformations. I'm tempted to add the given linear transformations like 2T + 2T - 1T - 2T which gives
T[ 1 3 ]
,,[ 4 2 ]
which I believe it is 10. Is that valid?
Correct. For linear transformations, T(aA+bB)=aT(A)+bT(B).

T[ a b ]
,,[ c d ]
Should be obvious by now.

#### war485

Wow, that's amazing that you saw through the pattern in < 15 minutes. It's overly dead obvious now. Thank you so very much!

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