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) =(4,-6), and T(e3)=(-5,3), where e₁,e2, e3 are the columns of the 3 x 3 identity matrix. a. Is the linear transformation one-to-one? ... CA. 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. C. T is not one-to-one because the columns of the standard matrix A are linearly independent. D. T is one-to-one because T(x) = 0 has only the trivial solution. b. Is the linear transformation onto? OA. T is not onto because the standard matrix A contains a row of zeros. B. T is not onto because the columns of the standard matrix A span R². C. T is onto because the standard matrix A does not have a pivot position for every row. 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) =(4,-6), and T(e3)=(-5,3), where e₁,e2, e3 are the columns of the 3 x 3 identity matrix. a. Is the linear transformation one-to-one? ... CA. 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. C. T is not one-to-one because the columns of the standard matrix A are linearly independent. D. T is one-to-one because T(x) = 0 has only the trivial solution. b. Is the linear transformation onto? OA. T is not onto because the standard matrix A contains a row of zeros. B. T is not onto because the columns of the standard matrix A span R². C. T is onto because the standard matrix A does not have a pivot position for every row. D. T is 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(e₁) = (1,3), T(e₂) = (4. – 6), and T(e3)= (-5,3), where e₁,e2, e3 are the columns of the 3 x 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.
C. T is not one-to-one because the columns of the standard matrix A are linearly independent.
OD. T is one-to-one because T(x) = 0 has only the trivial solution.
b. Is the linear transformation onto?
OA. T is not onto because the standard matrix A contains a row of zeros.
O.B. T is not onto because the columns of the standard matrix A span
R².
OC. T is onto because the standard matrix A does not have a pivot position for every row.
D. T is onto because the columns of the standard matrix A span
R².
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