# Matrix associated to a linear map

• Sep 11th 2009, 01:14 AM
math.dj
Matrix associated to a linear map
R^3 = L(direct sum)W,with dim(L)=1.Suppose T:R^3--->R^3 is a linear map.T(L)subset of L and T(W)subset of W.
Find a basis B of R^3 such that m(T;B) is a 3*3 matrix with entries...a(11) nonzero number, a(21) zero, a(31) zero, a(12) zero, a(22) nonzero number, a(32) nonzero number, a(13) zero, a(23) nonzero number, a(33) nonzero number..
• Sep 11th 2009, 08:46 AM
HallsofIvy
Quote:

Originally Posted by math.dj
R^3 = L(direct sum)W,with dim(L)=1.Suppose T:R^3--->R^3 is a linear map.T(L)subset of L and T(W)subset of W.
Find a basis B of R^3 such that m(T;B) is a 3*3 matrix with entries...a(11) nonzero number, a(21) zero, a(31) zero, a(12) zero, a(22) nonzero number, a(32) nonzero number, a(13) zero, a(23) nonzero number, a(33) nonzero number..

Choose non-zero vectors $v_1, v_2, v_3$ so that $v_1$ is in L (and since L has dimension 1, { $v_1$} is a basis for L) and { $v_2, v_3$} is a basis for W. You can do that because L directsum W= $R^3$. Applying T to $v_1$ you get $Tv_1= av_1+ 0v_2+ 0v_3$ because T maps L to itself. Applying T to either $v_2$ or $v_3$ gives $0v_1+ bv_2+ cv_3$ because T maps W to itself.
• Sep 11th 2009, 09:22 PM
math.dj
W is not mentioned which plane it is..so can i take any two vectors..(like (1,0,-1/2),(0,1,-3/4) )..n what is the use of the condition T(L) subset of L and T(W) subset of W..is it used to find the linear map as the map is not mentioned..n what happens when T(L) subset of W and T(W) subset of L..
• Sep 12th 2009, 04:45 AM
HallsofIvy
Quote:

Originally Posted by math.dj
W is not mentioned which plane it is..so can i take any two vectors..(like (1,0,-1/2),(0,1,-3/4) )..n what is the use of the condition T(L) subset of L and T(W) subset of W..is it used to find the linear map as the map is not mentioned..n what happens when T(L) subset of W and T(W) subset of L..

If you really have no idea what the question is asking (and it appears from this that you don't) the best thing you can do is go to your teacher and ask for more explanation.
• Sep 12th 2009, 06:30 AM
math.dj