Question: 1a) Let L : R n -> R m be a linear transformation. If Ker(L) = zero vector and Range(L) = R m , show
1a) Let L : Rn -> Rm be a linear transformation. If Ker(L) = zero vector and Range(L) = Rm , show that m = n.
1b) Let L : Rn -> Rm and M : Rm -> Rp be linear transformations. Show that Range(M composite L ) is a subset of Range(M)
1c) Let L : Rn -> Rm and M : Rm -> Rp be linear transformations. Show that Ker(L) is a subset of Ker(M composite L )
1d) Let L1,L2,L3 : -> R2 -> R2 be linear transformations defined by L1(x1,x2)=(x1-x2,x1+x2), L2(x1,x2)=(x1+3x2,x1+5x2), L3(x1,x2)=(x1+x2,2x1+3x2).
(i)Determine if {L1,L2,L3} is linearly dependent or independent. Note that the zero transformation
O : R2 -> R2 is the linear transformation that satisfies O(x1, x2)=(0,0) for any x1,x2 that is real. (Hint : this is no different than with vectors in Rn - you simply have linear transformations rather than vectors).
(ii)Let M : R2 -> R2 be defined by M(x1,x2)=(x1-x2,5x1+x2). Express M as a linear combination of L1,L2,L3, if possible. Show your work.
1e) Let L be a linear transformation from Rn -> Rn such thatL(x)=x where x is a vector and is in Rn.
(i)Prove that Ker(L) = {0}.
(ii)Explain why L is a one-to-one correspondence.
(iii)Show that the standard matrix of L must satisfy [L]T[L] = I. You may use the hint that (1st hint) the dot product of a vector a with a vector b is = aTb where a and b are vectors in Rn. (2nd hint) If A,B are symmetric matrices in Mn x n(R) such that the dot product of a vector x with Ax is equal to the dot product of vector x with Bx where x is in Rn, then A = B
Step by Step Solution
There are 3 Steps involved in it
Get step-by-step solutions from verified subject matter experts