Synthetic division, functions, interest

A geologist you spoke with is concerned about the rate of land erosion around the base of a dam. Another geologist is studying the magma activity within the earth in an area of New Zealand known for its volcanic activity. One of the shortcuts they apply when doing calculations in the field is to use synthetic division. After t ...continues

Velocity and Time Graphs

The Velocity-time graph below shows the motion of a cyclist: (a) Describe in words the motion of the cyclist? (b) Calculate the acceleration of the cyclist from: (i) A to B (ii) B to C (iii) C to D (c) If the cyclist travelled a total distance of 200m, calculate the total time for this motion.

Show that any matrix A can be written as a sum of rank-1 matrices. And show how these rank-1 matrices can be chosen so that only r of them are necessary (where r=rank(A)).

Show that any matrix A can be written as a sum of rank-1 matrices. And show how these rank-1 matrices can be chosen so that only r of them are necessary (where r=rank(A)).

Proof : Adjoints and Sturm-Liouville Theorem

1. Let’s define the operator M as follows: Mu = f(x) u’’ + g(x) u’ + h(x) u Now define the adjoint of M as M* and let M*v = (fv)’’ – (gv)’ + hv = fv’’ + (2f’ – g)v’ + (f’’ – g’ + h)v Show that (M*)* = M

Eigenvalues

Find the eigenvalues of the matrix: [c1 c2....cn] [c1 c2....cn] [c1 c2....cn] [...............] Please see the attached file for the fully formatted problem.

Eigenvalues

Find the eigenvalues of the following matrix [0 0 1 ] [1 0 w+1+1/w ] [0 1 -w-1-1/w ] where w = e^(2 pi i/3) Please see the attached file for the fully formatted problem.

Eigenvectors from Transformations : Reflection, Shear and Rotation

In each part find as many linearly independent eigenvectors as you can by inspection (by visualizing the effect of the transformation of R^2). For each of your eigenvectors, find the corresponding eigenvalue by inspection; then check your results by computing the eigenvalues and bases for the eigenspaces from the standard matrix ...continues

Eigenvalues and EigenVectors : Let A be a square matrix such that A^3 = A. What can you say about the eigenvalues of A?

Let A be a square matrix such that A^3 = A. What can you say about the eigenvalues of A?

Linear Algebra : Find Currents Given Kirchhoff's Circuit and Voltage Law Equations (KCL and KVL)

1. An electrical circuit is shown in Figure 1. Figure 1 The governing KVL and KCL equations are: v-R2i2-R4i4 = 0; -R2i2+R1i1+R3i3 = 0; -R4i4 -R3i3+R5i5 = 0 i6 = i1+i2; i2 +i3 = i4; i1 = i3+i5; i4+i5 = i6 Find the currents for the case: R1 = 1 k; R2 = 5 k; R3 = 2 k; R4 = 10 k; R5 ...continues

Linear Algebra : Calculating heat loss through a wall

Engineers use the concept of thermal resistance R to predict the rate of heat loss through a building wall in order to determine the heating system's requirements. This concept relates the heat flow rate q through a material to the temperature difference ∆T across the material: q = . This relation is like the voltage-curr ...continues