The eigenspace of \(T, \lambda\) is the set of all eigenvectors of \(T\) corresponding to \(\lambda\), plus the \(0\) vector.

constituents

• \(T \in \mathcal{L}(V)\)
• \(\lambda \in \mathbb{F}\), an eigenvalue of \(T\)

requirements

\begin{equation} E(\lambda, T) = \text{null}\ (T - \lambda I) \end{equation}

i.e. all vectors such that \((T- \lambda I) v = 0\).

where, \(E\) is an eigenspace of \(T\).

additional information

sum of eigenspaces is a direct sum

\(E(\lambda_{1}, T) + … + E(\lambda_{m}, T)\) is a direct sum.

see Extended Kalman Filter

The Elastic Modulus is a measurement of how much deformation takes place given some force on the system. Formally, it is the slope of the stress-strain curve, defined by:

\begin{equation} E = \frac{stress}{strain} \end{equation}

The units in pascals as it is: force per area (pascals) divided by deformation (dimensionless, as it is a fraction of old shape over new shape \(\frac{V}{V}=1\)).

Depending on how its measured, it is called different things:

• Young’s Modulus: tensile elasticity—tendency for object to deform along an axis with force applied (usually that is just called the Elastic Modulus)
• Shear’s Modulus: shear elasticity—tendency of an object to shear (deform in shape with the constant volume) with force applied
• Bulk Modulus: volumetric elasticity—tendency for an object to deform in all directions when uniformly loaded

Eleanor Roosevelt is the first lady of the US.

• Created minimum wage
• Wrote a weekly column named My Day, in 135 newspapers
• 2x a week broadcast

Though Coulomb’s Law allow us to calculate the force between any two individual charges, one can note that most of it is independent of the second test charge. In fact, each charge emits a field around itself of the shape:

\begin{equation} \vec{E( r)} = k \frac{q}{r^{2}} = \frac{1}{4\pi \epsilon_{0}} \frac{q}{r^{2}} \end{equation}

Or, you can think of it as moving a test charge \(q\) around the charge of interest, then calculating:

\begin{equation} \vec{E} = \frac{\vec{F_{e}}}{q} \end{equation}