21-241 Matrices and Linear Transformations
Homework 4
Due Wednesday, October 12th
1. Let A be a linear tranformation from a vector space X into a vector space Y and B be a linear
transformation from a vector space Y to a vector space Z . Dene the composit
21-241 Matrix Algebra
Homework 5 Solutions
Due Wednesday, October 19
23
1
1. Let A = 4 1 5.
1
(a) Find the projection matrix PA which projects R3 onto CS (A).
23
1
aaT
Solution: Let a = 4 1 5, so that A = [a]. Then PA = T =
aa
1
21 1 13
4
3
1
3
1
3
3
1
3
21-241 Matrices and Linear Transformations
Homework 6 Solutions
1. An (n n) symmetric matrix A is called positive denite if xT Ax > 0 for all x Rn , x = 0. Let A be
a (2 2) symmetric matrix. Prove the following.
If A is positive denite, then a11 > 0 and
21-241 Matrices and Linear Transformations
Homework 2
Due Friday, September 16
1. Use the denition of matrix multiplication involving summations, i.e.
n
(A)i,k (B )k,j
(AB )ij =
k=1
to prove that (AB )T = B T AT , where A is an m n matrix and B is an n p
21-241 Matrix Algebra
Homework 3
Due Friday, September 23
1. Let A be an m
n matrix.
(a) Prove that the column space of A, CS (A), is a subspace of Rm .
Solution: Let a1 ,a2 ,. . . ,an be the column vectors of A. The linear combination
0a1 + 0a2 +
+ 0an =
21-241 Matrices and Linear Transformations
Homework 4 Solutions
1. Let A be a linear tranformation from a vector space X into a vector space Y and B be a linear
transformation from a vector space Y to a vector space Z . Dene the composition of A and B as
21-241 Matrix Algebra
Homework 5
Due Wednesday, October 19
23
1
1. Let A = 4 1 5.
1
(a) Find the projection matrix PA which projects R3 onto CS (A).
(b) Find a basis for LN S (A).
(c) Let B be the matrix whose columns are the basis vectors found in part (
Carnegie Mellon University
Department of Mathematical Sciences
21-241 Matrix Algebra
Review for Exam 1
Your rst exam will be on Wednesday September 28th, during class. Please arrive a few minutes early so that
we may start on time. No calculators, notes,
21-241 Matrix Algebra
Homework 3
Due MONDAY, September 26
1. Let A be an m
n matrix.
(a) Prove that the column space of A, CS (A), is a subspace of Rm .
(b) Prove that the nullspace of A, N S (A), is a subspace of Rn .
2. Let M be the vector space of all
21-241 Matrices and Linear Transformations
Homework 2 Solutions
1. Use the denition of matrix multiplication involving summations, i.e.
n
(A)i,k (B )k,j
(AB )i,j =
k=1
to prove that (AB )T = B T AT , where A is an m n matrix and B is an n p matrix.
Soluti
Carnegie Mellon University
Department of Mathematical Sciences
21-241 Review Exam 3, Fall, 2011
Your exam shall consist of problems similar to the homework and problems done in class.
Monday, November 28th will be a review day. Please ask any questions yo
Carnegie Mellon University
Department of Mathematical Sciences
21-241 Matrix Algebra
Review for Final Exam
Your nal exam will be on Monday December 19th, 5:30 pm to 8:30 pm in McConomy Auditorium in the University
Center. Please arrive a few minutes early
Carnegie Mellon University
Department of Mathematical Sciences
21-241 Matrix Algebra
Review for Final Exam
Your nal exam will be on Monday December 19th, 5:30 pm to 8:30 pm in McConomy Auditorium in the University
Center. Please arrive a few minutes early
ml;
1. Let .4=i
[_.
[a] Determine the .4 = Li: factorization.
l
.1
1
IL:-
2 I z RL3KIJL 7' l _2 RadRuff: l
Ll a /"> O I o F> 0 l
: I o Zz'fnrz'wa o 2' 12 o 0
a z. I "L
L; l a a (/1: a , a
3 I t o a -z
I 2
[b] Use this factorization [1.:'. forward a
Homework 1 - Due Friday Sept. 9th
Attached Files:
HW1.pdf (41.104 KB)
Homework 1 consists of the problems in the attached pdf file. You will need to solve
these problems and turn them in at the beginning of lecture on Friday September
9th.
In addition, pl
21-241 Matrix Algebra
Homework 1 Solutions
1. Here we prove that matrix multiplication is associative. Use the denition of matrix multiplication
involving summations, i.e.
(AB )i;j =
n
X
Ai;k Bk;j ,
k=1
to prove that (AB ) C = A (BC ), where A is an m
mat
21-241 Matrix Algebra
Homework 1
Due Friday, September 9
1. Here we prove that matrix multiplication is associative. Use the denition of matrix multiplication
involving summations, i.e.
(AB )i;j =
n
X
(A)i;k (B )k;j ,
k=1
to prove that (AB ) C = A (BC ),
Carnegie Mellon University
Department of Mathematical Sciences
21-241 Matrix Algebra
Review for Exam 2
Exam 2 will be on Monday October 31th, during class. Please arrive a few minutes early so that we may start on
time. No calculators, notes, books, etc.,