Final answer:
To demonstrate that T is a linear transformation for a vector space of 2x2 matrices, we must verify that the properties of vector addition and scalar multiplication are preserved, applying analogous principles from vector operations in a rectangular coordinate system.
Step-by-step explanation:
To show that T is a linear transformation for the vector space V of 2×2 matrices, we need to verify two properties: The transformation must be closed under vector addition and scalar multiplication. As per the given reference material, for any vectors, say A and B, in V and a scalar k, the transformation T should satisfy T(A + B) = T(A) + T(B) and T(kA) = kT(A).
In the case of a vector à represented in a rectangular coordinate system by Axİ + AyĴ + Azâ, we see vector operations such as dot products and scalar multiplication being used. For example, to show A İ = Ax, one might take the dot product of vector à with the unit vector İ. Similar operations are used to demonstrate invariance under coordinate rotations. These principles can be analogously applied to matrices to demonstrate linearity, ensuring dimensional consistency during operations.