Discussion
Let T be a linear transformation from R2 to R2 such that :
\[ T\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix},\ T\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}. \]
Then, \(T\begin{bmatrix} 3 \\ 4 \end{bmatrix}\) :
Answer: C
The values determine T completely, and \(T(3, 4)^{T} = (5, 3)^{T}\).
The vectors (1, 0) and (1, 2) form a basis of R2, so a linear map is fixed by their images. Since \((0, 2) = (1, 2) - (1, 0)\):
\[ \begin{aligned} T(0, 2) &= (2, 1) - (1, 1) = (1, 0) \\ T(0, 1) &= (1/2, 0) \\ T(3, 4) &= 3T(1, 0) + 4T(0, 1) \\ &= (3, 3) + (2, 0) = (5, 3) \end{aligned} \]
As a check, \((3, 4) = 1 \cdot (1, 0) + 2 \cdot (1, 2)\), and \(1 \cdot (1, 1) + 2 \cdot (2, 1) = (5, 3)\). 'Cannot be determined' would be right only if the two given vectors were dependent. (3, 2) is \(T(1, 0) + T(1, 2)\), the image of (2, 2) rather than (3, 4), and (4, 5) is not the image of (3, 4) under any map satisfying both conditions.
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