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5. Use Theorem 1.8 .2 to show that \( T \) is NOT a matrix transformation. \[ T(x, y)=\left(x^{2}, y\right) \] Proof:

A) It violates the closure property of vector addition.
B) It violates the closure property of scalar multiplication.
C) It does not preserve the origin.
D) It is not a real-valued function.

User Deebee
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Final answer:

T(x, y) = (x^2, y) does not satisfy the properties of a matrix transformation because it violates the closure property of vector addition. This is shown by the fact that T applied to the sum of two vectors is not equal to the sum of the transformations of the vectors; therefore, option A is correct.

Step-by-step explanation:

To determine whether T given by T(x, y) = (x^2, y) is a matrix transformation, we need to check if it satisfies two main properties: the closure property of vector addition and the closure property of scalar multiplication. A transformation is a matrix transformation if it operates linearly on a vector space.

Let's test for the closure property of vector addition:

Consider two vectors v1 = (a, b) and v2 = (c, d). The transformation T would map these to T(v1) = (a^2, b) and T(v2) = (c^2, d). Adding these transformed vectors yields (a^2 + c^2, b + d). However, applying T to the sum of original vectors (a + c, b + d) we get T(a + c, b + d) = ((a + c)^2, b + d), which simplifies to (a^2 + 2ac + c^2, b + d).

Clearly, (a^2 + c^2, b + d) != (a^2 + 2ac + c^2, b + d). This demonstrates that T does not preserve vector addition, violating the closure property, and thus it is not a matrix transformation.

While the property of scalar multiplication is also required for matrix transformations, we have already identified that T fails to be a matrix transformation just based on vector addition. Therefore, option A is the correct reason.

User StuartLC
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