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(Solved): For this assignment, only refer to definition(s)/(t)heorem(s)/(r)esult(s)/(f)acts available in the t ...
For this assignment, only refer to definition(s)/(t)heorem(s)/(r)esult(s)/(f)acts available in the
textbook Linear algebra and Its Applications, 6th edition, by David C. Lay, Steven R. Lay, Judi
J. McDonald. Please quote the definition or theorem and its page number that you used (if
any) in your solution
The QRQR decompositionA with columns
v_(1)=[[0],[1],[1],[0],[0]],v_(2)=[[1],[0],[1],[-1],[1]],v_(3)=[[1],[4],[-1],[1],[-1]]
into the form QR by using the method in Section 6.4 of Lay's book.
(Refer to the rubric for marking scheme) The sets
A={a_(1),a_(2),a_(3),a_(4)}={[[1,0],[1,0]],[[0,1],[0,1]],[[1,-1],[0,1]],[[1,0],[1,1]]}
and
B={b_(1),b_(2),b_(3),b_(4)}={[[1,1],[1,1]],[[1,0],[0,-1]],[[0,0],[1,0]],[[0,1],[0,0]]}
are two ordered bases for the vector space M_(22)(R) of all 2\times 2 real matrices
over R.
Given that, for x=[[a,b],[c,d]] in M_(22)(R),
x=ca_(1)+(d+c-a)a_(2)+(d+c-b-a)a_(3)+(2a+b-2c-d)a_(4)
and
x=(1)/(2)(a+d)b_(1)+(1)/(2)(a-d)b_(2)-(1)/(2)(a-2c+d)b_(3)+(1)/(2)(a-2b+d)b_(4)
(a) Determine the change-of-coordinates matrix P_(BlarrA) from A to B.
[6 marks ]
(b) Suppose that T:M_(22)(R)->M_(22)(R) is defined by
T(x)=cb_(1)+(d+c-a)b_(2)+(d+c-b-a)b_(3)+(2a+b-2c-d)b_(4)
for any x in M_(22)(R). Show that T is a linear transformation.
[4 marks]
