MAT 1341B: Introduction to Linear Algebra
Instructor: Erhard Neher
Final Exam
(December 21, 2004)
Family Name:
First Name:
Student number:
Please read these instructions carefully:
This is a closed book exam, and no notes of any kind are allowed. The use
MAT 1341B: Introduction to Linear Algebra
Instructor: Erhard Neher
Midterm Exam
(October 21, 2004)
Family Name:
First Name:
Student number:
Please read these instructions carefully:
You have 80 minutes to complete this exam.
This is a closed book exam,
MAT2341,11/18
Diagonalizationofmatrix
If A, D similar , then matrixP:A=P1DP
Recall,similarmatricescorrespondtothe
samelinearoperator.
Note:IfA&Baresimilar,thenA(t)=B(t)
I=identitymatrix
LetA=P1BP,thenA(t)=B(t),because
Note:P1IP=P1P=I
P1tIP=t(P1IP)=tI
Defi
MAT2341, 11/01
Recall:
If you have F:U->V a linear map, U,V vector spaces over a field k, then f is called
nonsingular if kerF = cfw_0.
If F is nonsingular => to F is one to one.
F is an isomorphism if F is one-to-one and onto (linear map that is bijectiv
MAT2341, 11/04
Sidenote about the problems:
5.25: F:R2->R2 is F(x,y)=(x-y, x-2y)
Knowing it's nonsingular, find F-1
So, F-1(u,v) = (2u-v, u-v)
Example 5.7 p. 177
Having G:R3->R3 rotation v R3 about the z-axis through an angle .
So G(x,y,z)=(xCos - ySin, x
MAT2341, 11/11
Recall: Suppose you have V a K-vectorspace and S, S' bases of V
P = change of basis matrix from S to S'.
Theorem: v V, [v]S' = P-1[v]S
[v]S=P[v]S'
Proof: Assume dimV=3 (proof in higher dimension is similar).
S = cfw_u1,u2,u3
S' = cfw_v1,v2,