(Solved): The set of all 2\times 2 matrices of the form 9. [[a,1],[1,b]] with the standard matrix addition and ...
The set of all
2\times 2matrices of the form 9.
[[a,1],[1,b]]with the standard matrix addition and scalar multiplication. The set of all
2\times 2matrices of the form 10.
[[a,0],[0,b]]with the standard matrix addition and scalar multiplication. The set of all real-valued functions
fdefined everywhere on the real line and such that
f(1)=0, with the operations 11. defined in Example 4. The set of all
2\times 2matrices of the form 12.
[[a,a+b],[a+b,b]]with matrix addition and scalar multiplication. The set of all pairs of real numbers of the form
(1,x)with the operations 13.
(1,y)+(1,y^('))=(1,y+y^(')), and ,k(1,y)=(1,ky)The set of polynomials of the form
a+bxwith the operations 14.
(a_(0)+a_(1)x)+(b_(0)+b_(1)x)=(a_(0)+b_(0))+(a_(1)+b_(1))x,and
,k(a_(0)+a_(1)x)=(ka_(0))+(ka_(1))xThe set of all positive real numbers with the operations 15.
x+y=xy, and ,kx=x^(k)The set of all pairs of real numbers
(x,y)with the operations 16.
(x,y)+(x^('),y^('))=(\times ^('),yy^(')), and ,k(x,y)=(kx,ky)Show that the following sets with the given operations fail to be vector spaces by identifying all axioms that fail to hold. 17. (a) The set of all triples of real numbers with the standard vector addition but with scalar multiplication defined by
k(x,y,z)=(k^(2)x,k^(2)y,k^(2)z)(b) The set of all triples of real numbers with addition defined by
(x,y,z)+(u,v,w)=(z+w,y+v,x+u)and standard scalar multiplication. (c) The set of all
2\times 2invertible matrices with the standard matrix addition and scalar multiplication. 18. Show that the set of all
2\times 2matrices of the form
[[a,1],[1,b]]with addition defined by
[[a,1],[1,b]]+[[c,1],[1,d]]=[[a+c,1],[1,b+d]]and scalar multiplication defined by
k[[a,1],[1,b]]=[[ka,1],[1,kb]]is a vector space. What is the zero vector in this space?
