Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Practice Final
Name: Solutions
This is a (long) practice exam. The real exam will consist of 6 problems.
In the real exam, no calculators, electronic devices, books, or notes may be used.
Show yo
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Practice midterm 3
Name: Solutions
This is a practice exam. The real exam will consist of 4 problems.
In the real exam, no calculators, electronic devices, books, or notes may be used.
Show your
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Practice midterm 1
Name: Solutions
This is a practice exam. The real exam will consist of at most 4 problems.
In the real exam, no calculators, electronic devices, books, or notes may be used.
Sh
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Midterm 3, November 16
Name: Solutions
No calculators, electronic devices, books, or notes may be used.
Show your work. No credit for answers without justication.
Good luck!
1.
/10
2.
/10
3.
/10
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Midterm 2, October 19
Name: Solutions
No calculators, electronic devices, books, or notes may be used.
Show your work. No credit for answers without justication.
Good luck!
1.
/10
2.
/10
3.
/8
4.
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Midterm 1, September 21
Name: Solutions
No calculators, electronic devices, books, or notes may be used.
Show your work. No credit for answers without justication.
Good luck!
1.
/10
2.
/10
3.
/10
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Similar matrices
1
Change of basis
Consider an n n matrix A and think of it as the standard representation of a transformation
TA : Rn Rn . If we pick a dierent basis cfw_v1 , . . . , vn of Rn , wh
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Schur decomposition
Let us illustrate the algorithm to nd a Schur decomposition, as in 6.1, Theorem 1.1.
Example: Find a Schur decomposition of the matrix
A=
7 2
.
12 3
Solution: First, we want an e
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Orthogonal matrices and rotations
1
Planar rotations
Denition: A planar rotation in Rn is a linear map R : Rn Rn such that there is a plane
P Rn (through the origin) satisfying
R(P ) P and R|P = som
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Some properties of matrices
Let A be an m n matrix. We discuss two interesting properties that A can have, which will
be stated in several equivalent ways.
Property 1
Proposition: The following are
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Least squares solution
1. Curve tting
The least squares solution can be used to t certain functions through data points.
Example: Find the best t line through the points (1, 0), (2, 1), (3, 1).
Solu
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Gram-Schmidt orthogonalization
Let us illustrate the fact that the Gram-Schmidt orthogonalization process works in any inner
product space, not just Rn (or Cn ).
Example: Consider the real inner pro
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Preview of Gauss-Jordan elimination ( 2.2)
Consider the linear system
x1 x2
4x1 + 3x2
=3
=5
whose augmented matrix is
1 1 3
.
435
We solve the system using Gauss-Jordan elimination:
1 1 3
435
R2 4R1
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Working in coordinates
In these notes, we explain the idea of working in coordinates or coordinate-free, and how the
two are related.
1
Expressing vectors in coordinates
Let V be an n-dimensional ve
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Column space and null space
The following example illustrates the notion of dimension and culling down a linearly dependent collection of vectors.
Let
A = a1 a2 a3
1
2
=
3
1
2
4
6
2
1
1
.
1
1
Find t
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Homework 11 solutions
Section 6.2
2.13. (1 pt check) Let A be a normal operator on Cn (without loss of generality) whose
eigenvalues all have modulus 1. Since A is normal, it can be unitarily diagon
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Homework 10 solutions
Section 5.5
5.2. (1 pt check) Since A has rank 2, let us nd the orthogonal projections onto the
1-dimensional subspaces Null A and Null AT .
111
111
111
222
201
A = 1 3 2 0 2 1
Math 416 - Abstract Linear Algebra
Fall 2011, section E1
Homework 9 solutions
Section 5.2
2.2. (2 pts) Let x Rn and consider the equivalent conditions:
x (Col AT ) = (Row A)
x row i of A for all i = 1, . . . , m
(row 1 of A, x)
.
m
.
=0R
.
(row m of A, x