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96 Chapter 3
4.2 Projection of the Vector on the Plane Described
by Two Column Vectors of the Matrix ‘X’
Figure 3-5. Projection of the vector on the plane
The Linear combination of certain number of n-dimensional independent
vectors forms the vector space. Thus vector space is defined as the space
spanned by the set of independent vectors.
Consider the vector ‘a’ projected on the plane described by the vectors
‘x1’ and ‘x2’. Note that the matrix X =[x1 x2]. The vector ‘p’ can be
represented as the linear combination of the vector ‘x1’ and ‘x2’ (i.e)
column vectors of the matrix X as illustrated in the figure 3-4
p=u1*x1+u2*x2
p=[X] u1
u2
p=[X][s]
Also the error vector [a-p] is orthogonal to x1 and x2 as illustrated in the
figure.
T
T
⇒(a-Xs) x1 =0 and (a-Xs) x2 =0
T
Jointly represented as [a-Xs] X = 0
0
T
T
⇒a X =(Xs) X
