One more linear transformation

In summary, the given equations show that T is a linear transformation, with specific values for T[1 0] and T[1 1]. Using the property of linear transformations, we can find T[1 3] and T[a b].
  • #1
war485
92
0

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?
 
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  • #2
war485 said:
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
[tex]T\[
\left(
\begin{array}{cc}
0 & 0 \\
0 & 1 \end{array}
\right)=1\] [/tex]

Similarly you get

[tex]T\[
\left(
\begin{array}{cc}
0 & 0 \\
1 & 0 \end{array}
\right) = T
\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \end{array}
\right) = 1\][/tex]

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.
 
  • #3
Wow, that's amazing that you saw through the pattern in < 15 minutes. It's overly dead obvious now. Thank you so very much! :smile:
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps a set of inputs to a set of outputs, while preserving the properties of addition and scalar multiplication.

2. What is the purpose of a linear transformation?

Linear transformations are used to transform data in a way that simplifies analysis and problem solving. They are also used in many areas of science, such as physics and engineering, to model real-world scenarios.

3. How is a linear transformation different from other types of transformations?

Unlike other transformations, such as non-linear or affine transformations, linear transformations preserve the relative distances and angles between points. This means that parallel lines remain parallel after the transformation.

4. What are some common examples of linear transformations?

Some common examples of linear transformations include translation, rotation, scaling, and shearing. These transformations can be represented by matrices and are often used in computer graphics and image processing.

5. How are linear transformations used in data analysis?

In data analysis, linear transformations are used to transform data into a new coordinate system, making it easier to analyze and interpret. They can also be used to reduce the dimensionality of data, which can aid in the visualization and understanding of complex data sets.

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