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Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T: R³-R², T(e₁) = (1,3), T(e₂)=(2, -11), and T(e3)= (-3,8), where e₁, 2, 3 are the columns of the 3×3 identity matrix. a. Is the linear transformation one-to-one? A. T is not one-to-one because the standard matrix A has a free variable. B. T is one-to-one because the column vectors are not scalar multiples of each other. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is one-to-one because T(x) = 0 has only the trivial solution. b. Is the linear transformation onto? O A. T is not onto because the standard matrix A contains a row of zeros. O B. T is onto because the standard matrix A does not have a pivot position for every row. O C. T is not onto because the columns of the standard matrix A span R². O D. T is onto because the columns of the standard matrix A span R².

Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T: R³-R², T(e₁) = (1,3), T(e₂)=(2, -11), and T(e3)= (-3,8), where e₁, 2, 3 are the columns of the 3×3 identity matrix. a. Is the linear transformation one-to-one? A. T is not one-to-one because the standard matrix A has a free variable. B. T is one-to-one because the column vectors are not scalar multiples of each other. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is one-to-one because T(x) = 0 has only the trivial solution. b. Is the linear transformation onto? O A. T is not onto because the standard matrix A contains a row of zeros. O B. T is onto because the standard matrix A does not have a pivot position for every row. O C. T is not onto because the columns of the standard matrix A span R². O D. T is onto because the columns of the standard matrix A span R².

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T: R³-R², T(e₁) = (1,3), T(e₂) = (2, -11), and T(e3)= (-3,8), where e₁,e₂, e3 are the columns of the 3x3 identity matrix. a. Is the linear transformation one-to-one? A. T is not one-to-one because the standard matrix A has a free variable. B. T is one-to-one because the column vectors are not scalar multiples of each other. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is one-to-one because T(x) = 0 has only the trivial solution. b. Is the linear transformation onto? O A. T is not onto because the standard matrix A contains a row of zeros. O B. T is onto because the standard matrix A does not have a pivot position for every row. O C. T is not onto because the columns of the standard matrix A span R². O D. T is onto because the columns of the standard matrix A span R².
Transcribed Image Text:Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T: R³-R², T(e₁) = (1,3), T(e₂) = (2, -11), and T(e3)= (-3,8), where e₁,e₂, e3 are the columns of the 3x3 identity matrix. a. Is the linear transformation one-to-one? A. T is not one-to-one because the standard matrix A has a free variable. B. T is one-to-one because the column vectors are not scalar multiples of each other. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is one-to-one because T(x) = 0 has only the trivial solution. b. Is the linear transformation onto? O A. T is not onto because the standard matrix A contains a row of zeros. O B. T is onto because the standard matrix A does not have a pivot position for every row. O C. T is not onto because the columns of the standard matrix A span R². O D. T is onto because the columns of the standard matrix A span R².
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