Estimated Time: 25 minutes

Students answer two small white board questions and then carry out several calculations in matrix and bra-ket notation.

• Tabletop Whiteboard with markers
• A Finding Matrix Elements handout for each student

This activity is introduced with two small whiteboard questions

1. If $|v\rangle \doteq \pmatrix{v_x \\ v_y \\ v_z}$, what does $\langle1|v\rangle$?

Students generally find this activity quite easy and generate answers like $v_x$, the projection onto the $\langle1|$ direction, or the x component.

2. If $A\doteq \pmatrix{a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}}$, what does $A|1\rangle$ represent?

The purpose of this question is to connect this operation to the linear transformations activity and to get the students to see that the result of this calculation is a column vector which represents the result of the linear transformation A “operating on” the vector $|1\rangle$