Find the linear transformation T : ℝ2 → ℝ2 that has the values given below on the given basis.T 2 1 = 4 6, T 1 1 = 3 7(a) First write a general vector v as a linear combination of the basis vectors.v =v1 v2 = v1−v2 2 1 + 2v2−v1 1 1(b) This implies the following.T(v) = T v1 v2= 2v1 v1+6v2 = v
Question
Find the linear transformation T : ℝ2 → ℝ2 that has the values given below on the given basis.T 2 1 = 4 6, T 1 1 = 3 7(a) First write a general vector v as a linear combination of the basis vectors.v =v1 v2 = v1−v2 2 1 + 2v2−v1 1 1(b) This implies the following.T(v) = T v1 v2= 2v1 v1+6v2 = v
Solution
It seems like there's a bit of confusion in your question. However, I'll try to guide you through the process of finding a linear transformation given its action on a basis.
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First, we need to express a general vector v in ℝ2 as a linear combination of the basis vectors. Let's say our basis vectors are b1 = (2,1) and b2 = (1,1). Then any vector v in ℝ2 can be written as:
v = xb1 + yb2
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Given that T(b1) = (4,6) and T(b2) = (3,7), we can express the transformation T(v) as:
T(v) = xT(b1) + yT(b2)
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Substituting the given transformations of the basis vectors, we get:
T(v) = x*(4,6) + y*(3,7)
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Therefore, the linear transformation T : ℝ2 → ℝ2 is given by:
T(v) = (4x + 3y, 6x + 7y)
This transformation takes a vector v in ℝ2 and maps it to another vector in ℝ2. The coefficients x and y are the coordinates of the vector v with respect to the basis vectors b1 and b2.
Similar Questions
Find the linear transformation T : ℝ2 → ℝ2 that has the values given below on the given basis.T 2 1 = 4 6, T 1 1 = 3 7(a) First write a general vector v as a linear combination of the basis vectors.v =v1 v2 = v1−v2 2 1 + 2v2−v1 1 1(b) This implies the following.T(v) = T v1 v2= 2v1 v1+6v2 = v
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