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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 25 "Bases and Span with Maple
" }}{PARA 19 "" 0 "" {TEXT -1 27 "Math 230 March 10, 1998" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "This demo
nstrates how one can use Maple to work with bases." }}}{SECT 1 {PARA
3 "" 0 "" {TEXT -1 33 "Load the linear algebra package. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "This loads all the
routines you need for this demonatration." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "with(linalg):" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT
-1 15 "Finding a basis" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "First en
ter some vectors. Here there are four vectors in " }{XPPEDIT 18 0 "R^
4" "*$%\"RG\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0
98 "v1 := vector([1,2,1,2]);\nv2 := vector([2,1,-2,-1]);\nv3 := evalm(
v1 - v2);\nv4 := vector([0,1,1,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 12 "The command " }{HYPERLNK 17 "basis" 2 "linalg,basis" "" }{TEXT
-1 50 " finds a basis for the span of a set of vectors. " }}{PARA 0 "
" 0 "" {TEXT -1 61 "It returns a minimal set of vectors that has the s
ame span as" }}{PARA 0 "" 0 "" {TEXT -1 28 "the original set of vector
s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Not
e that the routine only returns the names of the vectors. To see the \+
vectors" }}{PARA 0 "" 0 "" {TEXT -1 28 "one must do additional work."
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "basis(\{v1,v2,v3,v4\});\nmap(eval
,\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "One can also use " }
{HYPERLNK 17 "basis" 2 "linalg,basis" "" }{TEXT -1 50 " to find a n or
dered basis using a list for input." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0
21 "basis([v1,v2,v3,v4]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "It i
s also easy to find a basis of the span of the set of vectors that are
" }}{PARA 0 "" 0 "" {TEXT -1 74 "orthogonal. This is sometimes very u
seful. One applies the Gramm-Schmidt" }}{PARA 0 "" 0 "" {TEXT -1 28 "
process through the command " }{HYPERLNK 17 "GrammSchmidt" 2 "linalg,G
ramSchmidt" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 51 "This is not a normalized set of orthogon
al vectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 27 "GramSchmidt([v1,v2,v3,v4]);" }}}}{SECT 1 {PARA 3 ""
0 "" {TEXT -1 20 "Working in a matrix." }}{PARA 0 "" 0 "" {TEXT -1 57
"One can also find a basis through the row or column space" }}{PARA 0
"" 0 "" {TEXT -1 11 "operations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "First construct a matrix whose row
s are the 4-vectors v1, v2, v3, and v4." }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "M1 := transpose(augment(v1,v2,
v3,v4));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "One can use the " }
{HYPERLNK 17 "rowspace" 2 "linalg,rowspace" "" }{TEXT -1 47 " command \+
to get a basis for the rowspace of M1." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rowspace(M1);" }}}{EXCHG {PARA
0 "" 0 "" {TEXT -1 69 "One can use the command to get another basis f
or the rowspace of M1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "rowspan(M
1);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Simple exercises." }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "(1) Find a basis for the span of
the vectors [1,2,1,2,1], [2,2,1,1,2], [3,1,2,1,3],\n[2,-1,0,-1,2], an
d [-2, 3, 2, 4, -2]. Which vector does the basis command" }}{PARA 0 "
" 0 "" {TEXT -1 79 "indicate is a linear combination of the others? R
eplace it with another vector" }}{PARA 0 "" 0 "" {TEXT -1 19 "to give \+
a basis of " }{XPPEDIT 18 0 "R^n" ")%\"RG%\"nG" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "(2) Find an orthogonal basis for \+
the span of the vectors in the problem (1)." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 24 "(3) Apply the commands " }{HYPERLNK 17 "gausselim" 2 "
linalg,gausselim" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "gaussjord" 2 "
linalg,gaussjord" "" }{TEXT -1 31 " to the matrix M1 from above. " }}
{PARA 0 "" 0 "" {TEXT -1 62 "How do you think Maple finds its bases fo
r the rowspace of M1?" }}}}}{MARK "1 0 0" 33 }{VIEWOPTS 1 1 0 1 1
1803 }