Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T: R³-R², T(e₁) = (1,4), T(e₂)=(3,-5), and T(3)=(-4,1), where e₁, 2, 3 are the columns of the 3x3 identity matrix. a. Is the linear transformation one-to-one? CE OA. T is one-to-one because the column vectors are not scalar multiples of each other. OB. T is not one-to-one because the columns of the standard matrix A are linearly independent. OC. T is one-to-one because T(x) = 0 has only the trivial solution. OD. T is not one-to-one because the standard matrix A has a free variable.
Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T: R³-R², T(e₁) = (1,4), T(e₂)=(3,-5), and T(3)=(-4,1), where e₁, 2, 3 are the columns of the 3x3 identity matrix. a. Is the linear transformation one-to-one? CE OA. T is one-to-one because the column vectors are not scalar multiples of each other. OB. T is not one-to-one because the columns of the standard matrix A are linearly independent. OC. T is one-to-one because T(x) = 0 has only the trivial solution. OD. T is not one-to-one because the standard matrix A has a free variable.
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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1.9 #7
Transcribed Image Text:Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer.
\( T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \), \( T(e_1) = (1, 4) \), \( T(e_2) = (3, -5) \), and \( T(e_3) = (-4, 1) \), where \( e_1, e_2, e_3 \) are the columns of the \( 3 \times 3 \) identity matrix.
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a. Is the linear transformation one-to-one?
- \( \circ \) A. T is one-to-one because the column vectors are not scalar multiples of each other.
- \( \circ \) B. T is not one-to-one because the columns of the standard matrix A are linearly independent.
- \( \circ \) C. T is one-to-one because \( T(x) = 0 \) has only the trivial solution.
- \( \circ \) D. T is not one-to-one because the standard matrix A has a free variable.
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