Question

If A is a 4 x 3 matrix, then the transformation x maps to Ax maps R³ onto R⁴. Choose the correct answer below.

A. True. The transformation maps R³ onto R⁴.
B. False. The dimensions of A do not allow this mapping.
C. True. The mapping is valid for any matrix size.
D. False. The transformation cannot map to higher dimensions.

Answer 1

Option (B), The transformation x maps to Ax with A being a 4 x 3 matrix cannot map R³ onto R⁴. The dimensions of A do not allow for every vector in R³ to have a unique image in R⁴, hence option B is correct.

Step-by-step explanation:

If A is a 4 x 3 matrix, then the transformation x maps to Ax cannot map R³ onto R⁴ because the matrix transformation from R³ to R⁴ requires a 4 x 3 matrix to perform a linear transformation from a 3-dimensional vector space to a 4-dimensional vector space.

This means that every vector in R³ is taken to a unique vector in R⁴. However, since there are only 3 columns in A (representing the input vector space of R³), there cannot be more unique output vectors in R⁴ than there are input vectors in R³.

Hence, the correct answer to the student's question is B. False. The dimensions of A do not allow this mapping.