Subspaces
Definition and Examples
Definition
Let V be a vector space and let S be a subset of V such that S is a vector space with
the same + and * from V. Then S is called a subspace of V.
Remark: Every vector space V contains at least two subspace, name
Change of Basis
Coordinates
Consider the vector v = (2,5,3) in R3. In writing these coordinates we mean
v = 2e1 + 5e2 + 3e3
Where
e1 = (1,0,0)
e2 = (0,1,0)
e3 = (0,0,1)
are the standard basis vectors. Sometimes we are interested in finding the coordinates
MATH 203 MIDTERM I
Part 1
Please work out each of the given problems without the use of a calculator. Credit will be based
on the steps that you show towards the final answer. Show your work.
Problem 1
Consider the matrix
A. Use the definition of the dete
Review of Some Linear Algebra
In this discussion, we expect some familiarity with matrices. For a review of the basics click
here. We will rely heavily on calculators and computers to work out the problems. Consider
some examples.
Example
Solve the system
Theory of Systems of Linear Differential Equations
It turns out that the theory of systems of linear differential equations resembles the theory of
higher order differential equations. This discussion will adopt the following notation. Consider
the system
Systems with Complex Eigenvalues
In the last section, we found that if
x' = Ax
is a homogeneous linear system of differential equations, and r is an eigenvalue with
eigenvector z, then
x = zert
is a solution. (Note that x and z are vectors.) In this discu
MATH 203
LINEAR ALGEBRA
Monday Wednesday and Friday 1:00 to 2:40 PM Room A211
Instructor: Larry Green
Phone Number
Office: 541-4660 Extension 341
Internet e-mail: [email protected]
Home Page: http:/www.ltcc.edu/academics.asp?
scatID=5&catID=34" http:
Math 203 Practice Midterm 2
Please work out each of the given problems. Credit will be based on the steps towards the
final answer. Show your work.
Problem 1
Let L: R2 -> R3 be a linear transformation such that
L (1,4) = (1,-1,3) and
L (0,2) = (2,1,4)
Fin
Math 203 Practice Final Exam Printable Key
Please work out all of the following problems. Credit will be given based on the progress that
you make towards the final solution. Show your work. No calculators allowed for this page.
Problem 1
Let
Find an orth
Math 203 Practice Midterm 3
Please work out each of the given problems. Credit will be based on the steps towards the
final answer. Show your work. Do your work on your own paper.
Problem 1
A physicist has plotted the position of a projectile over time. B
Orthonormal Bases in Rn
Orthonormal Bases
We all understand what it means to talk about the point (4,2,1) in R3. Implied in this notation is
that the coordinates are with respect to the standard basis (1,0,0), (0,1,0), and (0,0,1). We learn
that to sketch
Homogeneous Systems
In this discussion we will investigate how to solve certain homogeneous systems of linear
differential equations. We will also look at a sketch of the solutions.
Example
Consider the system of differential equations
x' = x + y
y' = -2x