An engineer? I had grown up among engineers, and I
could remember the engineers of the twenties very well indeed: their open,
shining intellects, their free and gentle humor, their agility and breadth of
thought, the ease with which they shifted from one engineering field to
another, and, for that matter, from technology to social concerns and art. Then
too, they personified good manners and delicacy of taste; well-bred speech that
flowed evenly and was free of uncultured words; one of them might play a musical
instrument, another dabble in painting; and their faces always bore a spiritual
imprint.
Aleksandr Solzhenitsyn, The Gulag
Archipelago
Math 251 Fall, 2009
Matrices, matrix operations
Linear systems, row reduced echelon form
Notes on linear independence and its connection to linear systems
The real definition of a linear function
The real definition of a vector space
These two sets of notes are an introduction to the way we define vector spaces and linear functions on vector spaces in abstract mathematics. There are many applications of these ideas to conventional vectors and to more general vectors spaces such as function spaces.
Practice problems involving matrices Answers
Problems from linear algebra text:
2.6 1 all; 2 all; 3 all; 4,6,7
3.6 1f, 2f, 3ef, 4ac, 6de
4.6 1 abd, 2 bcef, 3 bc, 7 ab, 9 a-d, 10, 11 cf
5.5 1 ehij, 2 dijl, 3, 6 cd, 7,8,15c,16c
Practice problems on linear systems Answers
The full expansion of a 3x3 determinant
(Note – this is not the “final” answer. Each remaining determinant with identical rows has value zero, and if the rows are distinct, you interchange them to get to the identity, multiplying by (-1) each time you interchange rows. The identity matrix has a determinant of 1, of course.)
Problems from calculus text:
10.1 2,3,4,5,6,7b,8,10,11,12,19,21,25,29
10.2 2,3,4,5,7,11,13,15,17
10.3 1,2,5,6,7,9,10, 14,15,22,25
10.4 2,5,7,8,9,15,23,27,28,29,30
10.5 3,5,7,9,10,11,13,15,17,21,23,25,27,31,35,39,,45,47,49,51
Given point P(-1,2,3) and plane 2x+2y-3z=3, find the intersection with the plane of the line through P that is normal to the plane.
Given point P(3,-1,2) find the point on the line x=1-2t,y=3+t,z=t that is closest to P.
Given a point P and a line r=r0+tv, explain how to find a vector in their common plane that is perpendicular to v.
10.6 1,3,5,11-23 odd
Practice not just plotting equations, but thinking of equations that produce a given desired type of surface and orientation.
10.7 5-10,13,15,17-22,23,33,39-43 odd,45,46,49,51
10.7 57,59,61,63
10.8 1,2,11,15,17,18,19,35,36
10.9 5,6,9,29,31
11.1 5,7,13,15,19,22,25,29
11.2 5,9,21
11. 3 2, 7-25,29,31,37,39,42,43,45,51,53
11.4 1,3,11,12,18,20,21,25,29
11.5 1,3,5,7,17,21,22,25,26,29,33
11.6 3,5,7,10,13,15,19,24,25,30
11.7 3,5,15
12.1 11,17,29
12.2 1,3,7,9,11,19,21,25,31,33,35,40,44
12.3 1-8,10,12,13,14,16,17,18,21,23,25
12.4 8,9,11,17,21
12.5 9,14,1718,19,17,29,31,32,35,39
12.6 17,19,21,23
12.7 5,7,8,9a,11,21,31,32
Regions in 3 dimensions via MATLAB
Wedge cut out of a sphere (there are two pauses that require keystrokes to continue)
Some partial derivatives in thermodynamics This is a page out of a thermodynamics textbook, illustrating some fancy footwork with partial derivatives.
Some problems involving lines and planes
If you have access to MATLAB, paste each section into the command window in turn, and hit return to see the corresponding surface. There is also a successive display of a hyperboloid of two sheets, a cone, and a hyperboloid of one sheet, layered inside each other with a semitransparent surface so you can see inside – hit return to go to the next plot. You can play with the zoom button and the rotation button to get a better look.
The function slicer.m can be saved into your MATLAB working directory. Then you can use it to “slice through” any 3 –dimensional plot in a figure window. In the command window, just type slicer(‘x’) to take slices down the x-axis, and similarly for the y and z axes.
Here’s a saddle, strategically chopped off on top and bottom:
[X,Y]=meshgrid(-3:.01:3,-4:.01:4);Z=X.^2-Y.^2;Z(abs(Z)>4)=NaN;
surf(X,Y,Z);shading flat
alpha(.7)
Curves and tangents (a MATLAB script)
Notes on resolution of acceleration
solutions (handwritten)