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Math Help - Linear Transformations: please help!

  1. #1
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    Linear Transformations: please help!

    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)
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  2. #2
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    Quote Originally Posted by tanelly View Post
    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)
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