Does there exist a linear transformation T : R2 → R4 such that Range(T ) ={(x1, x2, x3, x4) : x1 + x2 + x3 + x4 = 0
Question
Does there exist a linear transformation T : R2 → R4 such that
Range(T) = {(x_1, x_2, x_3, x_4) : x_1 + x_2 + x_3 + x_4 = 0}
Solution
Yes, there does exist a linear transformation T : R2 → R4 such that Range(T) = {(x1, x2, x3, x4) : x1 + x2 + x3 + x4 = 0}.
Here's a step-by-step explanation:
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Define the linear transformation T : R2 → R4 as T(x, y) = (x, y, -x, -y).
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For any (x, y) in R2, T(x, y) is in the set {(x1, x2, x3, x4) : x1 + x2 + x3 + x4 = 0} because x + y - x - y = 0.
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Therefore, the range of T is a subset of {(x1, x2, x3, x4) : x1 + x2 + x3 + x4 = 0}.
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Conversely, for any (x1, x2, x3, x4) in the set {(x1, x2, x3, x4) : x1 + x2 + x3 + x4 = 0}, we can find (x, y) in R2 such that T(x, y) = (x1, x2, x3, x4) by setting x = x1 and y = x2.
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Therefore, the set {(x1, x2, x3, x4) : x1 + x2 + x3 + x4 = 0} is a subset of the range of T.
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Since the range of T and the set {(x1, x2, x3, x4) : x1 + x2 + x3 + x4 = 0} are subsets of each other, they are equal.
So, the linear transformation T : R2 → R4 such that Range(T) = {(x1, x2, x3, x4) : x1 + x2 + x3 + x4 = 0} does exist.
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