One more linear transformation

[war485]

Homework Statement

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 ]

Homework Equations

none.

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\[<br /> \left(<br /> \begin{array}{cc}<br /> 0 & 0 \\<br /> 0 & 1 \end{array}<br /> \right)=1\]$$

Similarly you get

$$T\[<br /> \left(<br /> \begin{array}{cc}<br /> 0 & 0 \\<br /> 1 & 0 \end{array}<br /> \right) = T<br /> \left(<br /> \begin{array}{cc}<br /> 0 & 1 \\<br /> 0 & 0 \end{array}<br /> \right) = 1\]$$

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!