Linear Transformations: please help!

May 2010
4
0
chapel hill, nc
Let V and W be vector spaces. A function L, with domain V and range W us called a linear transformation from V to W iff

vectors denoted by bold

i) for every m and n in V, L(m+n)=L(m) + L(n)
ii) for every m in V and every scalar c, L(cm) = c(L(m))

In each of the following problems, determine whether or not the function L is really a linear transformation. You are to use i and ii to determine this.

a) V is the vector space of real numbers, as is W. L(x) = 3x-2, where r is a real number


b)V is the vector space of ordered pairs of real numbers. W is the vector space of real numbers. L ( (x,y ) ) = ax+by where a and b are fixed real numbers

c) V is the vector space of 2X2 matrices with real entries and W is the vector space of real numbers.
where L(matrix) = determinant of the matrix (i could not draw the matrix on this form)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Let V and W be vector spaces. A function L, with domain V and range W us called a linear transformation from V to W iff

vectors denoted by bold

i) for every m and n in V, L(m+n)=L(m) + L(n)
ii) for every m in V and every scalar c, L(cm) = c(L(m))

In each of the following problems, determine whether or not the function L is really a linear transformation. You are to use i and ii to determine this.

a) V is the vector space of real numbers, as is W. L(x) = 3x-2, where r is a real number.
You mean "where x is a real number". Is L(3)= L(1)+ L(2)?


b)V is the vector space of ordered pairs of real numbers. W is the vector space of real numbers. L ( (x,y ) ) = ax+by where a and b are fixed real numbers
So L((x,y)+ (u,v))= L(x+u, y+ v)= a(x+u)+ b(y+ v). Is that the same as L(x,y)+ L(u,v)= ax+ by+ au+ bv?

Is L(t(x,y))= L(tx,ty)= atx+ bty the same as tL(x,y)= t(ax+ by)?
c) V is the vector space of 2X2 matrices with real entries and W is the vector space of real numbers.
where L(matrix) = determinant of the matrix (i could not draw the matrix on this form)