1.7.23 Linear transformations

Condition: Given linear operators \( \varphi \) and \( \psi \) in the space \( V^{3} \). 1. Find the matrices of the operators \( \varphi, \psi \) and \( \varphi \cdot \psi \) in the basis \( i, j, k \). 2. Find the kernel and image of the operators \( \varphi \) and \( \psi \). In the case of a non-zero kernel, describe them with equations. 3. Find out whether there is an inverse operator for \( \varphi \cdot \psi \). If yes, then describe its geometric meaning; if not, indicate the reason. \( \varphi \)-rotation around the axis \( O Z \) by \( 90^{\circ}, \psi \)-orthogonal projection onto the plane \( x-y+z=0 \).