Step 1: Understanding the Concept:
We need to find the matrix representation of a linear transformation T with respect to a given basis B. The columns of this matrix are the coordinate vectors of the images of the basis vectors, written with respect to that same basis B.
Step 2: Key Formula or Approach:
Let the basis be \(B = \{\vec{b}_1, \vec{b}_2\}\). The matrix of T relative to B, denoted \([T]_B\), is given by:
\[ [T]_B = \begin{pmatrix} [T(\vec{b}_1)]_B & [T(\vec{b}_2)]_B \end{pmatrix} \]
where \([T(\vec{b}_i)]_B\) is the coordinate vector of \(T(\vec{b}_i)\) with respect to the basis B.
1. Apply the transformation T to each vector in the basis B.
2. Express each resulting image vector as a linear combination of the basis vectors in B.
3. The coefficients of these linear combinations form the columns of the matrix \([T]_B\).
Step 3: Detailed Explanation:
The basis is \(B = \{\vec{b}_1, \vec{b}_2\}\) where \(\vec{b}_1 = (0,1)\) and \(\vec{b}_2 = (1,0)\).
The transformation is \(T(x,y)=(x,0)\).
1. Transform the basis vectors:
- For the first basis vector \(\vec{b}_1 = (0,1)\):
\[ T(\vec{b}_1) = T(0,1) = (0,0) \]
- For the second basis vector \(\vec{b}_2 = (1,0)\):
\[ T(\vec{b}_2) = T(1,0) = (1,0) \]
2. Find the coordinates of the images with respect to B:
- For \(T(\vec{b}_1) = (0,0)\):
We need to find scalars \(c_1, c_2\) such that \((0,0) = c_1\vec{b}_1 + c_2\vec{b}_2 = c_1(0,1) + c_2(1,0)\).
This gives \((0,0) = (c_2, c_1)\), so \(c_1=0\) and \(c_2=0\).
The coordinate vector is \([T(\vec{b}_1)]_B = \begin{pmatrix} 0 0 \end{pmatrix}\). This will be the first column of our matrix.
- For \(T(\vec{b}_2) = (1,0)\):
We need to find scalars \(d_1, d_2\) such that \((1,0) = d_1\vec{b}_1 + d_2\vec{b}_2 = d_1(0,1) + d_2(1,0)\).
The vector \((1,0)\) is exactly our second basis vector \(\vec{b}_2\).
So, \((1,0) = 0 \cdot (0,1) + 1 \cdot (1,0) = 0\vec{b}_1 + 1\vec{b}_2\).
This gives \(d_1=0\) and \(d_2=1\).
The coordinate vector is \([T(\vec{b}_2)]_B = \begin{pmatrix} 0 1 \end{pmatrix}\). This will be the second column of our matrix.
Note on Discrepancy: My second column is \((0,1)\). Let me recheck.
\(T(1,0) = (1,0)\). We need to write this in the basis B.
\((1,0) = d_1(0,1) + d_2(1,0)\). The vector \((1,0)\) is \(\vec{b}_2\).
Wait, the basis is given as B=\(\{(0,1), (1,0)\}\). Let \(\vec{e}_1 = (0,1)\) and \(\vec{e}_2 = (1,0)\).
\(T(\vec{e}_1)=T(0,1)=(0,0) = 0\vec{e}_1 + 0\vec{e}_2\). First column is \(\begin{pmatrix}00\end{pmatrix}\).
\(T(\vec{e}_2)=T(1,0)=(1,0) = \vec{e}_2 = 0\vec{e}_1 + 1\vec{e}_2\). Second column is \(\begin{pmatrix}01\end{pmatrix}\).
Matrix is \(\begin{pmatrix} 0 & 0 0 & 1 \end{pmatrix}\). This matches option (B).
Let me re-read the provided solution which is (C).
Maybe the order of basis vectors in the matrix is different? The convention is \(j\)-th column is \(T(\vec{b}_j)\).
The marked answer is \(\begin{pmatrix} 0 & 1 0 & 0 \end{pmatrix}\).
This would mean the first column is \(\begin{pmatrix}00\end{pmatrix}\) and the second is \(\begin{pmatrix}10\end{pmatrix}\).
First column: \(T(\vec{b}_1)=(0,0)\), which is \(0\vec{b}_1+0\vec{b}_2\). Correct.
Second column: \(T(\vec{b}_2)\) has coordinates \(\begin{pmatrix}10\end{pmatrix}\). This means \(T(\vec{b}_2) = 1\vec{b}_1 + 0\vec{b}_2 = \vec{b}_1 = (0,1)\).
But we calculated \(T(\vec{b}_2) = T(1,0) = (1,0)\).
So \(T(\vec{b}_2)=(1,0)\), but the coordinate vector required for matrix (C) is \(\begin{pmatrix}10\end{pmatrix}\), meaning \(T(\vec{b}_2)\) should be \(\vec{b}_1=(0,1)\). This is a contradiction.
Let's try the standard basis \(\{(1,0),(0,1)\}\).
\(T(1,0)=(1,0)\), \(T(0,1)=(0,0)\). The matrix would be \(\begin{pmatrix} 1 & 0 0 & 0 \end{pmatrix}\).
There seems to be an error in the question or the provided answer key.
Let's check the basis order one more time. Basis is \(B = \{\vec{b}_1=(0,1), \vec{b}_2=(1,0)\}\).
Image of first basis vector: \(T(\vec{b}_1) = T(0,1) = (0,0)\).
Coordinates of image: \((0,0) = 0\vec{b}_1 + 0\vec{b}_2\). So first column is \(\begin{pmatrix} 0 0 \end{pmatrix}\).
Image of second basis vector: \(T(\vec{b}_2) = T(1,0) = (1,0)\).
Coordinates of image: \((1,0) = c_1\vec{b}_1 + c_2\vec{b}_2 = c_1(0,1) + c_2(1,0) = (c_2, c_1)\).
So \(c_2=1, c_1=0\). Second column is \(\begin{pmatrix} 0 1 \end{pmatrix}\).
Resulting matrix: \(\begin{pmatrix} 0 & 0 0 & 1 \end{pmatrix}\). This is option (B). The key is wrong.
Step 4: Final Answer:
The correct matrix representation is \(\begin{pmatrix} 0 & 0 0 & 1 \end{pmatrix}\), which is option (B). The provided answer key for option (C) is incorrect.