The Unique Linear Transformation T

To find the unique linear transformation T: R^2 -> R^2, we are given two conditions:

1. T(1,2) = (2,3)
2. 7(0,1) = (1,4)

Understanding Linear Transformations

A linear transformation maps vectors from one vector space to another in a linear manner. It preserves vector addition and scalar multiplication properties.

In this case, T maps vectors in R^2 (a 2-dimensional vector space) to vectors in R^2 itself.

Finding the Rule for T

We can represent T as a 2x2 matrix, where each column represents the image of the standard basis vectors in R^2.

Let's denote T as:
T = | a b |
| c d |

To find the rule for T, we need to determine the values of a, b, c, and d.

Using Condition 1

From condition 1, we know that T(1,2) = (2,3).

This gives us the following equations:
a(1) + c(2) = 2
b(1) + d(2) = 3

Simplifying these equations, we get:
a + 2c = 2 ...(1)
b + 2d = 3 ...(2)

Using Condition 2

From condition 2, we know that 7(0,1) = (1,4).

This gives us the following equations:
b(0) + d(1) = 1
b(7) + d(7) = 4

Simplifying these equations, we get:
d = 1 ...(3)
7b + 7d = 4 ...(4)

Substituting the value of d from equation (3) into equation (4), we get:
7b + 7(1) = 4
7b + 7 = 4
7b = 4 - 7
7b = -3
b = -3/7

Finding a, c, and d

Substituting the value of d from equation (3) into equation (2), we get:
(-3/7)(1) + 2(1) = 3
-3/7 + 2 = 3
-3/7 = 3 - 2
-3/7 = 1
-3 = 7

This equation is not possible, which means there is no solution for T.

Conclusion

Based on the given conditions, there is no unique linear transformation T: R^2 -> R^2 that satisfies both conditions simultaneously.