Which of the following is/are true? (A) Every linear transformation from R 2 to R 2 maps lines onto points or lines (B) Every surjective linear transformation from R 2 to R 2 maps lines onto lines (C) Every bijective linear transformation from R 2 to R 2 maps pairs of parallel lines to pairs of parallel lines (D) Every bijective linear transformation from R 2 to R 2 maps pairs of perpendicular lines to pairs of perpendicular lines
Question
Which of the following is/are true? (A) Every linear transformation from R 2 to R 2 maps lines onto points or lines (B) Every surjective linear transformation from R 2 to R 2 maps lines onto lines (C) Every bijective linear transformation from R 2 to R 2 maps pairs of parallel lines to pairs of parallel lines (D) Every bijective linear transformation from R 2 to R 2 maps pairs of perpendicular lines to pairs of perpendicular lines
Solution
(A) True. Every linear transformation from R2 to R2 maps lines onto points or lines. This is because linear transformations preserve the operations of vector addition and scalar multiplication, which are the operations that define lines in R2.
(B) True. Every surjective (onto) linear transformation from R2 to R2 maps lines onto lines. This is because a surjective linear transformation is a function that hits every point in the codomain at least once, and since lines in R2 are defined by points, every line in the domain must map to a line in the codomain.
(C) False. Not every bijective linear transformation from R2 to R2 maps pairs of parallel lines to pairs of parallel lines. This is because bijective transformations, while they preserve the one-to-one correspondence between points in the domain and codomain, do not necessarily preserve the property of parallelism.
(D) False. Not every bijective linear transformation from R2 to R2 maps pairs of perpendicular lines to pairs of perpendicular lines. This is because, similar to (C), bijective transformations do not necessarily preserve the property of perpendicularity.
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