null measurements

Examples of null measurements in the following topics:

• Null Measurements

  • Null measurements balance voltages so there is no current flowing through the measuring devices that would interfere with the measurement.
  • Null measurements balance voltages, so there is no current flowing through the measuring device and the circuit is unaltered.
  • A potentiometer is a null measurement device for measuring potentials (voltages).
  • A variety of bridge devicesare used to make null measurements in circuits .
  • The potentiometer is a null measurement device.

• Kelvin Scale

  • The kelvin is a unit of measurement for temperature; the null point of the Kelvin scale is absolute zero, the lowest possible temperature.
  • The kelvin is a unit of measurement for temperature.
  • The Kelvin scale is an absolute, thermodynamic temperature scale using absolute zero as its null point.
  • The choice of absolute zero as null point for the Kelvin scale is logical.
  • The kelvin is the primary unit of measurement in the physical sciences, but it is often used in conjunction with the degree Celsius, which has the same magnitude.

• Absolute Temperature

  • Thermodynamic temperature is the absolute measure of temperature.
  • Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point ("absolute zero") is the temperature at which the particle constituents of matter have minimal motion and cannot become any colder.

• A Geometrical Picture

  • Any vector in the null space of a matrix, must be orthogonal to all the rows (since each component of the matrix dotted into the vector is zero).
  • Therefore all the elements in the null space are orthogonal to all the elements in the row space.
  • In mathematical terminology, the null space and the row space are orthogonal complements of one another.
  • Similarly, vectors in the left null space of a matrix are orthogonal to all the columns of this matrix.
  • This means that the left null space of a matrix is the orthogonal complement of the column $\mathbf{R}^{n}$ .

• Four-Dimensional Space-Time

  • Both observers measure how far the beam has traveled at each point in time and how long it took to travel to travel that distance.
  • That is to say, observer A measures:
  • where, for example, $\Delta t = t - t_0$; t is the time at which the measurement took place; and t0 is the time at which the light was turned on.
  • Events that are time-like or null do not share this property, and therefore there is a causal ordering between time-like events.
  • The reason is that if two space-time points are time-like or null separated, one can always send a light signal from one point to another.

• Back to Maxwell's Equations

  • Michelson and Morley attempted to measure the motion of the Earth through the aether, but failed.
  • Lorentz proposed that to understand the null result of the experiment objects moving through the aether contract by γ−1 = $\sqrt{1-v^2/c^2}$where γ is the Lorentz factor.

• Introduction to Least Squares

  • Often we make repeated measurements which, because of noise, for example, are not exactly consistent.
  • Suppose we make $n$ measurements of some quantity $x$.
  • Let $x_i$ denote the ith measurement.
  • The orthogonal complement of the column space is the left null space, so $A\mathbf{x_{ls}} - \mathbf{y}$ must get mapped into zero by $A^T$ :
  • $A^TA$ has the same null space as $A$$A\mathbf{x}=0$ .

• Examples

  • The null space of $A$ can be obtained by solving the system
  • So the null space is is the line spanned by
  • Find the row-reduced form and the null space of the matrix
  • The only element in the null space is the zero vector.
  • This means that the null space is spanned $(-2,1,1)$ .

• Eigenvectors and Orthogonal Projections

  • On the other hand, if we only include the terms in the sum associated with the $r$ nonzero singular values, then we get a projection operator onto the non-null space (which is the row space).
  • is a projection operator onto the null space.
  • This says that any vector in can be written in terms of its component in the null space and its component in the row space of .
  • $\mathbf{x} = I \mathbf{x} = \left(V_r V_r ^T + V_0 V_0 ^T \right) \mathbf{x} = (\mathbf{x})_{\rm row} + (\mathbf{x})_{\rm null}

• Absolute Zero

  • Therefore, it is a natural choice as the null point for a temperature unit system.
  • Explain why absolute zero is a natural choice as the null point for a temperature unit system