Problem $3$: A simple truss system In this problem we will analyze the truss structure below. All members are of equal length, $\backslash ($ L $\backslash ).$ A vertical force of magnitude F is applied at the bottom central node as shown below. The cross sectional area of the beams is $\backslash ($ A $\backslash )$ and the elastic modulus is $\backslash ($ E $\backslash ).$

$1.$ Write the full force vector with all known values and variables for the unknown values.

$2.$ Write the displacement vector with all known values and variables for the unknown values.

$3.$ What is the size of the global stiffness matrix?

$4.$ What is the size of the reduced global stiffness matrix after the displacement constraints are removed?

$5.$ By hand write the stiffness matrix for the labeled element A$,$ in the elemental coordinates.

$6.$ What are the rows and columns that element A contributes to in the global stiffness matrix?