Determine whether the statement below is true or false. Justify the answer. If A is a 3x2 matrix, then the transformation xAx cannot be one-to-one. Choose the correct answer below. O A. The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If Ax=b does not have a free variable, then the transformation represented by A is one-to-one. B. The statement is true. Transformations which have standard matrices which are not square cannot be one-to-one nor onto because they do not have pivot positions in every row and column. O C. The statement is true. A transformation is one-to-one only if the columns of A are linearly independent and a 3x2 matrix cannot have linearly independent columns. O D. The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. It does not matter what dimensions a vector is as long as it meets this requirement. The matrix A could be 1x4 and still represent a one-to-one transformation.

Determine whether the statement below is true or false. Justify the answer. If A is a 3x2 matrix, then the transformation xAx cannot be one-to-one. Choose the correct answer below. O A. The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If Ax=b does not have a free variable, then the transformation represented by A is one-to-one. B. The statement is true. Transformations which have standard matrices which are not square cannot be one-to-one nor onto because they do not have pivot positions in every row and column. O C. The statement is true. A transformation is one-to-one only if the columns of A are linearly independent and a 3x2 matrix cannot have linearly independent columns. O D. The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. It does not matter what dimensions a vector is as long as it meets this requirement. The matrix A could be 1x4 and still represent a one-to-one transformation.

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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Question

Determine whether the statement below is true or false. Justify the answer.
If A is a 3x2 matrix, then the transformation X+Ax cannot be one-to-one.
Choose the correct answer below.
O A. The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the
domain. If Ax=b does not have a free variable, then the transformation represented by A is one-to-one.
B. The statement is true. Transformations which have standard matrices which are not square cannot be one-to-one nor onto
because they do not have pivot positions in every row and column.
O C. The statement is true. A transformation is one-to-one only if the columns of A are linearly independent and a 3x2 matrix cannot
have linearly independent columns.
D. The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the
domain. It does not matter what dimensions a vector is as long as it meets this requirement. The matrix A could be 1x4 and still
represent a one-to-one transformation.

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The given statement, "If $A$ is a $3 \times 2$ matrix, then the transformation $x \mapsto A x$ cannot be one-to-one.

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