Question

Find a linear map [T:R²→R² ] so that T((1,2)=(2,3) and T(0,1)=(1,4)

Answer 1

To find the linear map T:R²→R² that takes (1,2) to (2,3) and (0,1) to (1,4), we express T as a 2x2 matrix. By solving the corresponding matrix equations, we find the matrix A = [[0, 1], [-5, 4]], which gives us the linear map T((x, y)) = (y, -5x + 4y).

Step-by-step explanation:

To find a linear map T:R²→R² such that T((1,2))=(2,3) and T((0,1))=(1,4), we can assume that the linear map T can be represented by a 2x2 matrix A = [[a, b], [c, d]]. The action of T on a vector (x, y) in R² is represented as a matrix-vector multiplication A*(x, y), resulting in a new vector in R².

Using the given mappings, we have two matrix equations:

A*(1, 2) = (2, 3)

1. The first equation corresponds to the system:

a*1 + b*2 = 2

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• Simplifying, we have:

a + 2b = 2

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• From the second equation, we get:

a*0 + b*1 = 1

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• Which simplifies to:

b = 1

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• Substituting b and d back into the first system:

a + 2*1 = 2 → a = 0

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• Thus, the matrix A representing the linear map T is:

A = [[0, 1], [-5, 4]]

This gives us the linear map T as follows:

T((x, y)) = (0*x + 1*y, -5*x + 4*y)