Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. T(X1 X2 X3) = (x₁ - 4x2 +6x3, X2-9x3) (a) Is the linear transformation one-to-one? O A. T is one-to-one because the column vectors are not scalar multiples of each other. B. T is one-to-one because T(x) = 0 has only the trivial solution. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is not one-to-one because the columns of the standard matrix A are linearly dependent.
Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. T(X1 X2 X3) = (x₁ - 4x2 +6x3, X2-9x3) (a) Is the linear transformation one-to-one? O A. T is one-to-one because the column vectors are not scalar multiples of each other. B. T is one-to-one because T(x) = 0 has only the trivial solution. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is not one-to-one because the columns of the standard matrix A are linearly dependent.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer.**
\[ T(x_1, x_2, x_3) = (x_1 - 4x_2 + 6x_3, \, x_2 - 9x_3) \]
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**(a) Is the linear transformation one-to-one?**
- **A.** T is one-to-one because the column vectors are not scalar multiples of each other.
- **B.** T is one-to-one because T(x) = 0 has only the trivial solution.
- **C.** T is not one-to-one because the columns of the standard matrix A are linearly independent.
- **D.** T is not one-to-one because the columns of the standard matrix A are linearly dependent.
---
To determine if the transformation is one-to-one, consider if:
- The transformation maps distinct inputs to distinct outputs.
- The columns of the standard matrix of the transformation are linearly independent, which would imply that T(x) = 0 has only the trivial solution.
Evaluate the choices accordingly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46029127-96a5-4dc2-997a-3c090d1aab2d%2Fa0ce529e-5def-4730-b724-5fe257265f01%2F8lkanls_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer.**
\[ T(x_1, x_2, x_3) = (x_1 - 4x_2 + 6x_3, \, x_2 - 9x_3) \]
---
**(a) Is the linear transformation one-to-one?**
- **A.** T is one-to-one because the column vectors are not scalar multiples of each other.
- **B.** T is one-to-one because T(x) = 0 has only the trivial solution.
- **C.** T is not one-to-one because the columns of the standard matrix A are linearly independent.
- **D.** T is not one-to-one because the columns of the standard matrix A are linearly dependent.
---
To determine if the transformation is one-to-one, consider if:
- The transformation maps distinct inputs to distinct outputs.
- The columns of the standard matrix of the transformation are linearly independent, which would imply that T(x) = 0 has only the trivial solution.
Evaluate the choices accordingly.
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