MATH 310
Homework due 06/22/2010
1. Let e1,e2,e3be the standard basis vectors in R3and consider the
ordered basis:
[e2,−e1,e3+e1]
Verify that this is actually a basis and find the coordinates of the
vector (1,1,1)Twith respect to that basis.
2. Let Tbe the linear map from R2to Rdefined by:
T((x, y)T) = x−y
Find its matrix (with respect to the standard bases) and find a basis
for its kernel.
3. Let Tbe the linear map from R3to R3defined by the formula:
T((x, y, z)T) = (3x+y+z , 2x−z, y)T
Find the matrix of Twith respect to the standard basis of R3.
4. Consider the subspace Uof R3defined by:
U={(x, y, z)T: 2x−y+z= 0, x +y= 0}
Express Uas the kernel of an appropriately defined linear map and
find the matrix of that map with respect to the standard bases of the
corresponding Rn’s.
5. Find a spanning vector set for the image of the linear map from R2to
R3defined by:
T((x, y)T) = (2x−y, x +y, y)T
1