ColumnSpace - Maple Help

LinearAlgebra

 RowSpace
 return a basis for the row space of a Matrix
 ColumnSpace
 return a basis for the column space of a Matrix

 Calling Sequence RowSpace(A, options) ColumnSpace(A, options)

Parameters

 A - Matrix options - (optional); constructor options for the result object

Description

 • The RowSpace(A) (ColumnSpace(A)) function returns a list of row (column) Vectors that form a basis for the Vector space spanned by the rows (columns) of Matrix A.  The Vectors are returned in canonical form with leading entries 1.
 • The row space (column space) of a zero Matrix is the empty list.
 • The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Vector constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order). If constructor options are specified in the calling sequence, each resulting Vector has the same specified options.
 • This function is part of the LinearAlgebra package, and so it can be used in the form RowSpace(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[RowSpace](..).

Examples

 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $A≔⟨⟨1,2,0⟩|⟨0,2,6⟩|⟨0,0,4⟩|⟨0,0,0⟩⟩$
 ${A}{≔}\left[\begin{array}{cccc}{1}& {0}& {0}& {0}\\ {2}& {2}& {0}& {0}\\ {0}& {6}& {4}& {0}\end{array}\right]$ (1)
 > $\mathrm{RowSpace}\left(A\right)$
 $\left[\left[\begin{array}{cccc}{1}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {1}& {0}\end{array}\right]\right]$ (2)
 > $\mathrm{ColumnSpace}\left(A\right)$
 $\left[\left[\begin{array}{c}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\end{array}\right]\right]$ (3)
 > $\mathrm{RowSpace}\left(⟨⟨0,0⟩|⟨0,0⟩⟩\right)$
 $\left[\right]$ (4)
 > $B≔⟨⟨x,0⟩|⟨y,1⟩⟩$
 ${B}{≔}\left[\begin{array}{cc}{x}& {y}\\ {0}& {1}\end{array}\right]$ (5)
 > $\mathrm{ColumnSpace}\left(B\right)$
 $\left[\left[\begin{array}{c}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\end{array}\right]\right]$ (6)
 > $C≔⟨⟨\frac{15}{4},-\frac{3}{4}\mathrm{sqrt}\left(10\right),\frac{1}{4}\mathrm{sqrt}\left(165\right)⟩|⟨-\frac{3}{4}\mathrm{sqrt}\left(10\right),\frac{3}{2},-\frac{1}{4}\mathrm{sqrt}\left(66\right)⟩|⟨\frac{1}{4}\mathrm{sqrt}\left(165\right),-\frac{1}{4}\mathrm{sqrt}\left(66\right),\frac{11}{4}⟩⟩$
 ${C}{≔}\left[\begin{array}{ccc}\frac{{15}}{{4}}& {-}\frac{{3}{}\sqrt{{10}}}{{4}}& \frac{\sqrt{{165}}}{{4}}\\ {-}\frac{{3}{}\sqrt{{10}}}{{4}}& \frac{{3}}{{2}}& {-}\frac{\sqrt{{66}}}{{4}}\\ \frac{\sqrt{{165}}}{{4}}& {-}\frac{\sqrt{{66}}}{{4}}& \frac{{11}}{{4}}\end{array}\right]$ (7)
 > $\mathrm{Normalizer}≔\mathrm{radnormal}$
 ${\mathrm{Normalizer}}{≔}{\mathrm{radnormal}}$ (8)
 > $\mathrm{ColumnSpace}\left(C\right)$
 $\left[\left[\begin{array}{c}{1}\\ {-}\frac{\sqrt{{10}}}{{5}}\\ \frac{\sqrt{{165}}}{{15}}\end{array}\right]\right]$ (9)