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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![Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer.
\[ T: \mathbb{R}^3 \rightarrow \mathbb{R}^2, \]
\[ T(e_1) = (1,3), \]
\[ T(e_2) = (4,-6), \]
\[ T(e_3) = (-5,3), \]
where \( e_1, e_2, e_3 \) are the columns of the \( 3 \times 3 \) 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.
- **D.** \( T \) is one-to-one because \( T(\mathbf{x}) = 0 \) has only the trivial solution.
b. Is the linear transformation onto?
- **A.** \( T \) 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 \(\mathbb{R}^2\).
- **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 \(\mathbb{R}^2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ec5dee8-4b81-4c1e-b6ca-023c2626018b%2F7944314f-27fe-4ddd-9f58-f147e71ded7a%2Fuv5gl0n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer.
\[ T: \mathbb{R}^3 \rightarrow \mathbb{R}^2, \]
\[ T(e_1) = (1,3), \]
\[ T(e_2) = (4,-6), \]
\[ T(e_3) = (-5,3), \]
where \( e_1, e_2, e_3 \) are the columns of the \( 3 \times 3 \) 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.
- **D.** \( T \) is one-to-one because \( T(\mathbf{x}) = 0 \) has only the trivial solution.
b. Is the linear transformation onto?
- **A.** \( T \) 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 \(\mathbb{R}^2\).
- **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 \(\mathbb{R}^2\).
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