Which of the following transformations
𝑇
:

𝑅
2
→
𝑅
2
is not linear?

Question

Which of the following transformations 
𝑇
:
 
𝑅
2
→
𝑅
2
 is not linear?

BlackTom AI · Accepted Answer

We start by clearly restating the question and listing all answer options so we can compare them.
Question: Which of the following transformations T: R^2 → R^2 is not linear?
Options:
1) T([x y]) = [0 0]
2) T([x y]) = [ x −3 2 + y ]
3) T([x y]) = [ 3y 2x ]
4) All of the above transformations are linear

Now, evaluate each option in terms of linearity, which requires two properties: additivity (T(u+v) = T(u) + T(v)) and homogeneity (T(cu) = cT(u)).
Option 1: T([x y]) = [0 0]. This is the zero transformation, sending every input to the zero vector. The zero map satisfies T(u+v) = 0 = 0+0 = T(u) + T(v) and T(cu) = 0 = c·0 = cT(u). Therefore, option 1 is linear.
Option 2: T([x y]) = [ x −3 2 + y ]. Interpreting the expression carefully, the output components appear to be: first component is x minus 3, and second component is 2 plus y. In other words, T([x y]) = [ x − 3; 2 + y ]. Such a mapping involves constant terms (−3 and +2) that do not depend linearly on x and y. A quick check shows T([0 0]) = [−3; 2], which is not the zero vector, violating the standard requirement for a linear map to map the origin to the origin. Consequently, additivity fails here, so this option is not linear.
Option 3: T([x y]) = [ 3y 2x ]. This can be written as T([x y]) = [0·x + 3·y; 2·x + 0·y], which corresponds to the matrix [[0, 3], [2, 0]] applied to [x; y]. Since it satisfies T(u+v) and T(cu) through standard matrix multiplication, this transformation is linear.
Option 4: All of the above transformations are linear. Given the analysis above, at least one of the previous options (specifically option 2) is not linear. Therefore, this statement is false because not every listed transformation is linear.
In summary, the assessment shows that the zero transformation (option 1) is linear, option 3 is linear, option 2 is not linear due to constant terms and origin mapping, and option 4 is false since not all options are linear.

Which of the following transformations 𝑇 : 𝑅 2 → 𝑅 2 is not linear?单项选择题

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