Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T: R3-R2, T(e,) = (1,3), T(e,) = (3, - 8), and T(ea) = (-4,5), where e,, e,, ez are the columns of the 3x3 identity matrix. ... a. Is the linear transformation one-to-one? O A. Tis one-to-one because T(x) = 0 has only the trivial solution. O B. Tis one-to-one because the column vectors are not scalar multiples of each other. O C. Tis not one-to-one because the columns of the standard matrix A are linearly independent. O D. Tis not one-to-one because the standard matrix A has a free variable. b. Is the linear transformation onto? O A. Tis not onto because the standard matrix A contains a row of zeros. O B. Tis onto because the standard matrix A does not have a pivot position for every row. OC. Tis onto because the columns of the standard matrix A span R2. O D. Tis not 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(e,) = (1,3), T(e,) = (3, - 8), and T(ea) = (-4,5), where e,, e,, ez are the columns of the 3x3 identity matrix. ... a. Is the linear transformation one-to-one? O A. Tis one-to-one because T(x) = 0 has only the trivial solution. O B. Tis one-to-one because the column vectors are not scalar multiples of each other. O C. Tis not one-to-one because the columns of the standard matrix A are linearly independent. O D. Tis not one-to-one because the standard matrix A has a free variable. b. Is the linear transformation onto? O A. Tis not onto because the standard matrix A contains a row of zeros. O B. Tis onto because the standard matrix A does not have a pivot position for every row. OC. Tis onto because the columns of the standard matrix A span R2. O D. Tis not onto because the columns of the standard matrix A span R2.
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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Transcribed Image Text:Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer.
T: R³→R2, T(e,) = (1,3), T(e2) = (3, - 8), and T(e3)=(- 4,5), where e,, e2, ez are the columns of the 3x3 identity matrix.
a. Is the linear transformation one-to-one?
O A. Tis one-to-one because T(x) = 0 has only the trivial solution.
B. Tis one-to-one because the column vectors are not scalar multiples of each other.
O C. Tis not one-to-one because the columns of the standard matrix A are linearly independent.
O D. Tis not one-to-one because the standard matrix A has a free variable.
b. Is the linear transformation onto?
O A. Tis not onto because the standard matrix A contains a row of zeros.
O B. Tis onto because the standard matrix A does not have a pivot position for every row.
O C. Tis onto because the columns of the standard matrix A span R2.
O D. Tis not onto because the columns of the standard matrix A span R².
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