CMS/ACM 104: Homework #2 Solutions
John J. Bruer (jbruer@cms.caltech.edu)
EXERCISES
1. (ST).
(a) First, we notice that range ST = cfw_Su : u range T . Therefore, range ST range S , and
so rank(ST ) rank S . Now let cfw_u 1, . . . , u m be a basis for ran

CMS/ACM 104: Homework #3 Solutions
John J. Bruer (jbruer@cms.caltech.edu) & Shaun Maguire
EXERCISES
1. (Change of Heart).
(a) We claim that RP = IV . Indeed, the definitions of P and R tell us that for any v =
RPv = R (P
n
X
j v j ) = R
j =1
n
X
j Pv j =

CMS/ACM 104: Homework #5 Solutions
John J. Bruer (jbruer@cms.caltech.edu)
EXERCISES
1. (The Annihilator). Let V be an n -dimensional inner-product space, and let W V be an m dimensional subspace. We can find an orthonormal basis (w 1, . . . , w m ) for W

CMS/ACM 104: Homework #1 Solutions
John J. Bruer (jbruer@cms.caltech.edu)
EXERCISES
1. (Piazza, Piazza). Do it if you havent already.
2. (Linear Spaces in Applied Math).
(a) We have already seen that the set of real-valued functions on R form a vector spa

CMS/ACM 104: Homework #4 Solutions
John J. Bruer (jbruer@cms.caltech.edu)
EXERCISES
1. (Invariance).
(a) The matrix of T with respect to the basis (x 1, . . . , x m , y 1, . . . , y p ) will be block diagonal. We
can see this because T (X ) X , and so the

Problem Set 4
CMS 104, Fall 2015
Assigned 19 October 2015. Due to Steele House by 4 pm on 27 October 2015.
This assignment covers eigenvalues, eigenvectors, generalized eigenvectors, nilpotents, diagonalization, triangularization, the spectral theorem, Jo

Problem Set 1
CMS/ACM 104, Fall 2015
Assigned 29 September 2015. Due to Steele House by 4 pm on 6 October 2015.
Homework Directions
As Caesar wrote, the problem set is divided into three parts: Exercises, Problems, and Applications.
Exercises are simple p

Problem Set 3
CMS/ACM 104, Fall 2014
Assigned 12 October 2015. Due to Steele House by 4 pm on 20 October 2015.
This assignment covers matrix representations, change of basis, eigenvalues, eigenvectors, diagonalization, and triangularization. Same rules as

Problem Set 5
CMS 104, Fall 2015
Assigned 2 November 2015. Due to Steele House by 4 pm on 10 November 2015.
This assignment covers inner products, orthogonality, the GramSchmidt process, orthogonal projection
and least squares, linear functionals, adjoint

Problem Set 2
CMS/ACM 104, Fall 2015
Assigned 5 October 2015. Due to Steele House by 4 pm on 13 October 2015.
This assignment covers linear transformations, composition, range, null space, the ranknullity
theorem, invertibility, isomorphism, linear system