Final Answer:
The statement "Every matrix transformation is a linear transformation" is False. (Option 2)
Step-by-step explanation:
Not every matrix transformation qualifies as a linear transformation. For a transformation to be linear, it must satisfy two essential properties: additivity and homogeneity. While matrix transformations often exhibit linearity, it is not universally true for all cases.
Consider a matrix transformation defined by the matrix A. If A is multiplied by a scalar c and then applied to a vector, it may not preserve both additivity and homogeneity. In other words, the resulting transformation might not adhere to the principles of linearity. Therefore, the statement is false in the general sense.
Linear transformations have specific criteria that matrices must meet to ensure linearity. These criteria involve matrix operations that guarantee the preservation of vector addition and scalar multiplication properties. While many matrix transformations are linear, it is crucial to verify these conditions for each case to ascertain their linearity.(Option 2)