The action of a linear transformation
𝑇
on a basis of
𝑅
2
is given.

Find the matrix of
𝑇
.

Question

The action of a linear transformation 
𝑇
 on a basis of 
𝑅
2
 is given.

Find the matrix of 
𝑇
.

BlackTom AI · Accepted Answer

The problem provides a linear transformation T on R^2 and specifies its action on a basis of R^2. We need to reconstruct the matrix of T from the given information and compare it against the provided answer options.

Option-by-option analysis:

Option A: The proposed matrix is presented as a 3-by-2 structure in the formatting shown. When forming the matrix of a linear transformation with respect to the standard basis, the columns should be the coordinates of T applied to the basis vectors (expressed in the standard basis). Since the basis in the statement is a pair of 2D vectors (likely [1,1]^T and [1,-1]^T), the columns of the matrix should each be a 2D vector. A 3-row column layout would be inconsistent with a linear map R^2 -> R^2, because the output should be 2-dimensional, not 3-dimensional. This discrepancy strongly suggests Option A is not the correct matrix, due to a mismatch in row count with the codomain.

Option B: This option also attempts to present a 2-by-2 matrix (two columns, two rows) with entries that would correspond to T(e1) and T(e2) in the standard basis if the inputs were the standard basis vectors. If the problem provides T on a basis other than the standard one, you would still convert those images into coordinates relative to the standard basis and place them as columns. If the numbers in Option B do not align with the given outputs for T on the specified basis vectors, then this option misrepresents the action of T. Without matching the exact given images, this option is unlikely to be correct.

Option C: This option also appears as a 2-by-2 matrix with a specific second column and a possibly altered first column. The key check is whether the first column equals T([1,1]^T) expressed in the standard basis and the second column equals T([1,-1]^T) in the standard basis. If either column deviates from the stated outputs, the entire matrix would be incorrect. Since the problem requires precise correspondence to the two basis vectors, any mismatch immediately disqualifies this option.

Option D: The final option seems to present another 2-by-2 matrix with different entries, including a negative entry in the bottom-right corner. The same principle applies: the columns must be exactly the standard-coordinate images of the two basis vectors given in the prompt. If this option’s columns do not match the provided T images for the two basis vectors, it is not the correct matrix. Additionally, the presence of unusual signs or zeros in places that contradict the given outputs would signal an error.

Synthesis and consistency check:
- The correct matrix of T, with respect to the standard basis, should have as its first column the coordinates of T applied to the first basis vector in the given basis (in standard coordinates), and as its second column the coordinates of T applied to the second basis vector (in standard coordinates).
- Since the problem statement specifies T on a basis of R^2, we must ensure both columns are 2-dimensional vectors. Any option proposing more than 2 rows or fewer than 2 rows immediately conflicts with the codomain being R^2.
- Each option must be checked by verifying that its columns match the given images T(v1) and T(v2) where v1 and v2 are the two basis vectors provided in the prompt. A mismatch in either column indicates an incorrect matrix.

In summary, the critical checks are: (1) do the columns have exactly two entries each, (2) do the two columns equal the standard-coordinate images of the two basis vectors as specified, and (3) do the column vectors align with the outputs shown for T([1;1]) and T([1;-1]) (or the exact basis presented in the problem)? If an option fails any of these checks, it is not the correct matrix. If it passes all checks, it is the correct matrix.

Note on structure and conventions:
- The reasoning above emphasizes matching outputs to the exact basis given, rather than assuming the standard basis by default.
- When evaluating similar problems, always recompute the column vectors explicitly from the given T(v1) and T(v2) and then compare against each candidate matrix column-by-column.
- If the problem had misformatted data or ambiguous outputs (as sometimes happens in the transcription of matrices), the safest approach is to reconstruct the two 2D output vectors from the provided T images and then test each candidate against those exact vectors.

The action of a linear transformation 𝑇 on a basis of 𝑅 2 is given. Find the matrix of 𝑇 .单项选择题

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