Question

If \(T:\mathbb{R}^2\to\mathbb{R}^2\) is a linear transformation defined by \[ T(x,y)=(x,0), \] then the matrix of \(T\) relative to the basis \[ B=\{(0,1),(1,0)\} \] is \(____\).

• A. \(\begin{pmatrix}1& 0\\1& 0\end{pmatrix}\)
• B. \(\begin{pmatrix}0& 0\\0& 1\end{pmatrix}\)
• C. \(\begin{pmatrix}0& 1\\0& 0\end{pmatrix}\)
• D. \(\begin{pmatrix}0& 0\\0& 0\end{pmatrix}\)

Hint:

For a matrix relative to a basis, transform each basis vector and write the resulting coordinate vectors as columns.

Answer 1

The Correct Option is B

Concept:
To find the matrix of a linear transformation relative to a basis, we apply the transformation to each basis vector and write the result as coordinates in the same basis.

Step 1: Write the given basis.
The given basis is: \[ B=\{(0,1),(1,0)\} \] Let: \[ b_1=(0,1) \] and \[ b_2=(1,0) \]

Step 2: Apply \(T\) on the first basis vector.
Given: \[ T(x,y)=(x,0) \] So: \[ T(b_1)=T(0,1) \] \[ T(0,1)=(0,0) \] Now write \((0,0)\) in terms of \(b_1\) and \(b_2\): \[ (0,0)=0b_1+0b_2 \] So the coordinate column is: \[ [T(b_1)]_B= \begin{pmatrix} 0 0 \end{pmatrix} \]

Step 3: Apply \(T\) on the second basis vector.
\[ T(b_2)=T(1,0) \] \[ T(1,0)=(1,0) \] But: \[ (1,0)=0(0,1)+1(1,0) \] So: \[ [T(b_2)]_B= \begin{pmatrix} 0 1 \end{pmatrix} \]

Step 4: Form the matrix.
The matrix of \(T\) relative to basis \(B\) has these coordinate vectors as columns: \[ [T]_B= \begin{pmatrix} 0 & 0 0 & 1 \end{pmatrix} \] \[ \therefore \text{Correct Answer is (B)} \]