The text you provided seems to be a mix of English and mathematical notation, but it's a bit unclear. However, I'll try to explain the process of finding a linear transformation based on the given values.

1. We are given that T(2,1) = (4,6) and T(1,1) = (3,7). These are the images of the basis vectors under the transformation T.

2. A general vector v in ℝ2 can be written as a linear combination of the basis vectors. So, we can write v = a(2,1) + b(1,1) for some scalars a and b.

3. To find the image of v under the transformation T, we apply T to the linear combination. Because T is a linear transformation, we can distribute it over the sum and factor it out of the scalar multiples. So, T(v) = T(a(2,1) + b(1,1)) = aT(2,1) + bT(1,1).

4. Substituting the given values of T(2,1) and T(1,1), we get T(v) = a(4,6) + b(3,7) = (4a + 3b, 6a + 7b).

So, the linear transformation T : ℝ2 → ℝ2 that has the given values on the given basis is T(v) = (4a + 3b, 6a + 7b), where v = a(2,1) + b(1,1).

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The text you provided seems to be a mix of English and mathematical notation, but it's a bit unclear. However, I'll try to explain the process of finding a linear transformation based on the given values.

1. We are given that T(2,1) = (4,6) and T(1,1) = (3,7). These are the images of the basis vectors under the transformation T.

2. A general vector v in ℝ2 can be written as a linear combination of the basis vectors. So, we can write v = a(2,1) + b(1,1) for some scalars a and b.

3. To find the image of v under the transformation T, we apply T to the linear combination. Because T is a linear transformation, we can distribute it over the sum and factor it out of the scalar multiples. So, T(v) = T(a(2,1) + b(1,1)) = aT(2,1) + bT(1,1).

4. Substituting the given values of T(2,1) and T(1,1), we get T(v) = a(4,6) + b(3,7) = (4a + 3b, 6a + 7b).

So, the linear transformation T : ℝ2 → ℝ2 that has the given values on the given basis is T(v) = (4a + 3b, 6a + 7b), where v = a(2,1) + b(1,1).

Knowee AI · Accepted Answer

The text you provided seems to be a mix of English and mathematical notation, but it's a bit unclear. However, I'll try to explain the process of finding a linear transformation based on the given values.

1. We are given that T(2,1) = (4,6) and T(1,1) = (3,7). These are the images of the basis vectors under the transformation T.

2. A general vector v in ℝ2 can be written as a linear combination of the basis vectors. So, we can write v = a(2,1) + b(1,1) for some scalars a and b.

3. To find the image of v under the transformation T, we apply T to the linear combination. Because T is a linear transformation, we can distribute it over the sum and factor it out of the scalar multiples. So, T(v) = T(a(2,1) + b(1,1)) = aT(2,1) + bT(1,1).

4. Substituting the given values of T(2,1) and T(1,1), we get T(v) = a(4,6) + b(3,7) = (4a + 3b, 6a + 7b).

So, the linear transformation T : ℝ2 → ℝ2 that has the given values on the given basis is T(v) = (4a + 3b, 6a + 7b), where v = a(2,1) + b(1,1).

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