Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
ShowthatdetAbCoAo = CIC2C3+CIC3C4+CIC2C4+C2C3C4. Find also detAICIA I in the fixed-free example.

Introduction to Linear Algebra 4
Invert AIc1A1 in the fixed-free example by multiplying AIIC!1 (AD-I.

Introduction to Linear Algebra 4
In the free-free case when ATCA in equation (9) is singular, add the three equations ATCAu = 1 to show that we need II + 12 + h = 0. Find a solution to ATCAu = 1 when the forces 1 = (-1,0,1) balance themselves. Find all solutions!

Introduction to Linear Algebra 4
Both end conditio~s for the free-free differential equation are du/ dx = 0: d (dU) . - dx c(x) dx = f(x) wIth du dx = at both ends. Integrate both sides to show that the force f(x) must balance itself, f f(x) dx = 0, or there is no solution. The complete solution is one particular solution u (x) plu

Introduction to Linear Algebra 4
In the fixed-free problem, the matrix A is square and invertible. We can solve AT y = 1 separately from Au = e. Do the same for the differential equation: dy Solve - dx = [(x) with y(1) = 0. Graph y(x) if [(x) = 1.

Introduction to Linear Algebra 4
The 3 by 3 matrix K 1 = AT CIA 1 in equation (6) splits into three "element matrices" CI EI + C2E2 + C3E3. Write down those pieces, one for each c. Show how they come from column times row multiplication of AT CIA 1. This is how finite element stiffness matrices are actually assembled.

Introduction to Linear Algebra 4
For five springs and four masses with both ends fixed, what are the matrices A and C and K? With C = I solve Ku = ones(4)

Introduction to Linear Algebra 4
Compare the solution u = (UI' U2, U3, U4) in Problem 7 to the solution of the continuous problem -u" = I with u(O) = 0 and u(l) = O. The parabola u(x) should correspond at x = t, ~, %, ~ to u-is there a (.6.xf factor to account for?

Introduction to Linear Algebra 4
Solve the fixed-free problem -u" = mg with u(O) = 0 and u'(l) = O. Compare u (x) at x = ~, ~, ~ with the vector u = (3mg, 5mg, 6mg) in Example 2.

Introduction to Linear Algebra 4
Suppose CI = C2 = C3 = C4 = 1, ml = 2 and m2 = m3 = 1. Solve ATCA u = (2,1,1) for this fixed-fixed line of springs. Which mass moves the most (largest u)?

Introduction to Linear Algebra 4
(MATLAB) Find the displacements u (l), ... , u (l00) of 100 masses connected by springs all with C = 1. Each force is I(i) = .01. Print graphs of u with fixed-fixed and fixed-free ends. Note that diag(ones(n, 1), d) is a matrix with n ones along diagonal d. This print command will graph a vector u: p

Introduction to Linear Algebra 4
(MATLAB) Chemical engineering has a first derivative dujdx from fluid velocity as well as d 2ujdx2 from diffusion. Replace dujdx by aforward difference, then a centered difference, then a backward difference, with .6.x = i. Graph your three numerical solutions of d 2u du - dx 2 + 10 dx = 1 with u(O)