Let \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be a linear transformation defined by \( T(x, y, z) = (2x + y - 3z, x - y + z, 3x + 2y - 4z) \). Find the kernel of \( T \).
Answer
Step 3: The solution set of the system of equations is the kernel of the transformation \( T \), so we obtain \( \text{Ker}(T) = \left\{ (x, y, z) \in \mathbb{R}^3 : x = \frac{2}{3}z, y = - \frac{1}{3}z \right\} \).
Step 1 ：Step 1: Set up the equation \( T(x, y, z) = 0 \), which gives us the system of equations: \begin{align*} 2x + y - 3z &= 0 \\ x - y + z &= 0 \\ 3x + 2y - 4z &= 0 \\ \end{align*}
Step 2 ：Step 2: Solve the system of equations. We can use the method of substitution or elimination. By adding the first two equations, we get \( 3x = 2z \), or \( x = \frac{2}{3}z \). Substituting this into the third equation, we get \( y = - \frac{1}{3}z \).
Step 3 ：Step 3: The solution set of the system of equations is the kernel of the transformation \( T \), so we obtain \( \text{Ker}(T) = \left\{ (x, y, z) \in \mathbb{R}^3 : x = \frac{2}{3}z, y = - \frac{1}{3}z \right\} \).
