Homework 1
MATH 120
Prof. Janet Vassilev
October 3, 2007
1. Find the transformation matrix Tfrom the basis B={(1,0,0)T,(1,1,0)T,(1,1,1)T}
to the standard basis.
2. Give the matrix representing the linear transformation
L(x, y, z) = (2x−3y , x + 4z, 2y−5z)
with respect to the standard basis.
3. Give the matrix representing the linear transformation
L(x, y, z) = (2x−3y , x + 4z, 2y−5z)
with respect to the basis (1,0,0)T,(1,1,0)T,(1,1,1)T.
4. Find the eigenvalues and their associated eigenvectors of the matrix
1 2 −1
1 0 1
4−4 5
.
5. Find an orthogonal matrix which diagonalizes µ1 2
2 1¶.
6. Find the orthogonal complement of the subspace W=span{(1,2,0,1)T,(1,0,3,2)T}.
7. Find a symmetric matrix which represents the quadratic form
f(x, y, z) = x2+ 4xy −xz + 6yz −3z2.
8. Determine if the matrix
1 2 −1
2 5 3
−1 3 5
is positive definite, negative definite or nei-
ther.
9. Problem 3.5 on page 35.
10. Problem 3.13 on page 36.
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