My office is Room 115 in Whitney Hall. My office hours will be MWF 1:00 - 2:00 and by appointment. To arrange for an office hour appointment, send an email at least 2 hours ahead of time to my email address: alex@math.binghamton.edu

All class meetings will all be held in person in Whitney Hall, Room G002. Section 5 is scheduled for MWF 2:50 - 4:20 PM, from August 23 through December 9, 2022.

The textbook for this course is below. Other references which can be helpful are listed on the main Math 304 webpage, but the lectures in my section will come from my own notes which are available as pdf files below.

``Linear Algebra" by Jim Hefferon, available for free download from the following link: ``Linear Algebra" by Jim Hefferon.

Homework assignments will come from the textbook, as described on the main Math 304 webpage.

All the material in my notes, and in the textbook will be covered if time permits. A list of major topics which should be covered is given at the end of this page.

The six sections of this course will be run separately by each instructor. This page contains details relevant only to my Section 5, but some general advice probably applies to everyone. There will be ten quizzes administered in class for 20% of your course grade. Homework done and evaluated through an online system will count for 5% of your course grade. Three 90-minute exams will be administered on announced dates during a normal class time. There will be one Final Exam (2 hours long) during the scheduled Finals period. Each 90-minute exam will be worth 15%, and the Final Exam will be worth 30%. The material being tested in each exam will be determined and announced approximately one week before the exam. The Final Exam will be comprehensive, covering the whole course. ANYONE WITH A SCHEDULING CONFLICT AFFECTING THE FINAL EXAM MUST CONTACT THE INSTRUCTOR AHEAD OF TIME TO ARRANGE FOR AN ALTERNATIVE TIME. A MESSAGE CAN BE LEFT AT THE MATH DEPT OFFICE (777-2147). NO ONE SHOULD MISS THE FINAL!

Information about the general scheduling of exams is posted on the main Math 304 page, but detailed information for this section will be posted here.

Exam 1 for Section 5 will be on Friday, Sept. 30, 2022.

Exam 2 for Section 5 will be on Friday, Nov. 4, 2022.

Exam 3 for Section 5 will be on Monday, Dec. 5, 2022.

**All students who are not in quarantine because of covid or other serious illness are
expected to take the exam in person during the scheduled class time. Students with an accommodation
for extra time must talk to Prof. Feingold ahead of time to make an arrangement. Students who are in
quarantine because of covid or other serious illness may ask to take the exam online using Gradescope.
**

After each Exam is graded and returned, solutions will be posted below, along with a letter grade interpretation of the numerical score.

The following link takes you to a pdf file with Exam 1_and_its solutions. The next link takes you to a pdf file showing the distribution of scores (0 - 75) on Exam 1, and the letter grade interpretation of the numerical scores. Exam 1_grade distribution.

The following link takes you to a pdf file with Exam 2_and_its solutions. The next link takes you to a pdf file showing the distribution of scores (0 - 75) on Exam 2, and the letter grade interpretation of the numerical scores. Exam 2_grade distribution.

**Any student with a special problem or a
finals conflict must contact the instructor (me) as soon as possible to make arrangements.**

Practice problems of various kinds may be posted on the main Math 304 webpage.

It is hoped that classes and exams will be held in-person, with whatever precautions are required due to COVID. Properly worn masks will probably not be required in Fall 2022. A student may test positive for COVID and have to be under quarantine, or a student may be too ill to take an exam or quiz (the flu is still a threat). To deal with those situations, I will make use of Gradescope so that students with a valid medical excuse can take an exam or quiz online remotely. That method of testing is not as secure as an in-person exam, so I have to rely on your honesty not to cheat in that situation. If I find that any cheating has occured, the penalty will be severe. Information about how to use Gradescope is in the next section.

In case the university administration cancels classes because of a snow emergency, we can still have a virtual class meeting on zoom. Panopto recordings of lectures from a previous semester are already available with the posted lecture notes, but if we have a class meeting on zoom, it will be recorded and a link made available on this webpage. Since snow cancelations are rare, I will not set up a recurring zoom meeting for this class, but I will only do one at a time if needed. An email announcing each zoom meeting would be sent in the morning including a link to the meeting.

The following link takes you to a pdf file with Quiz 1_and_its solutions.

The following link takes you to a pdf file with Quiz 2_and_its solutions.

The following link takes you to a pdf file with Quiz 3_and_its solutions.

The following link takes you to a pdf file with Quiz 4_and_its solutions.

The following link takes you to a pdf file with Quiz 5_and_its solutions.

The following link takes you to a pdf file with Quiz 6_and_its solutions.

The following link takes you to a pdf file with Quiz 7_and_its solutions.

The following link takes you to a pdf file with Quiz 8_and_its solutions.

The following link takes you to a pdf file with Quiz 9_and_its solutions.

The following link takes you to a pdf file with Quiz 10_and_its solutions.

Students who are unable to attend class in-person for exams or quizzes should find the exam or quiz on Gradescope, take it within the time limit, and submit the solutions as a single electronic pdf file. I highly recommend using CamScanner for this purpose. It is available from this website: CamScanner Website.

**All submitted documents should be sent to me using Gradescope according to the directions in
the following file:**
Gradescope Submitting Guide.

Here is the webpage for direct access to Gradescope: Gradescope login page.

Feb 12: Lecture recording and notes: Math 304-6 Lecture Notes pages 1-16 and Panopto recording of Lecture 1 on Feb. 12, 2021.

Feb 15: Lecture recording and notes: Math 304-6 Lecture Notes pages 17-26 and Panopto recording of Lecture 2 on Feb. 15, 2021.

Feb 17: Lecture recording and notes (note that page 36 has been modified on 2-22-2021): Math 304-6 Lecture Notes pages 27-36 and Panopto recording of Lecture 3 on Feb. 17, 2021.

Feb 19: Lecture recording and notes: Math 304-6 Lecture Notes pages 37-47 and Panopto recording of Lecture 4 on Feb. 19, 2021.

Feb 22: Lecture recording and notes: Math 304-6 Lecture Notes pages 48-61 and Panopto recording of Lecture 5 on Feb. 22, 2021.

Feb 24: Lecture recording and notes: Math 304-6 Lecture Notes pages 62-75 and Panopto recording of Lecture 6 on Feb. 24, 2021.

Feb 26: Lecture recording and notes: Math 304-6 Lecture Notes pages 76-85 and Panopto recording of Lecture 7 on Feb. 26, 2021.

Mar 1: Lecture recording and notes: Math 304-6 Lecture Notes pages 86-100 and Panopto recording of Lecture 8 on Mar. 1, 2021.

Mar 3: Lecture recording and notes: Math 304-6 Lecture Notes pages 101-106_Plus3 and Panopto recording of Lecture 9 on Mar. 3, 2021.

Mar 5: Lecture recording and notes: Math 304-6 Lecture Notes pages 107-118 and Panopto recording of Lecture 10 on Mar. 5, 2021.

Mar 8: Lecture recording and notes: Math 304-6 Lecture Notes pages 119-127 and Panopto recording of Lecture 11 on Mar. 8, 2021.

Mar 10: Lecture recording and notes: Math 304-6 Lecture Notes pages 128-134_Plus5 and Panopto recording of Lecture 12 on Mar. 10, 2021.

Mar 12: Lecture recording and notes: Math 304-6 Lecture Notes Exam1_Review and Panopto recording of Lecture 13 on Mar. 12, 2021.

Mar 19: Lecture recording and notes: Math 304-6 Lecture Notes 134.1-138.3.pdf and Panopto recording of Lecture 14 on Mar. 19, 2021.

Mar 22: Lecture recording and notes: Math 304-6 Lecture Notes 138.4-138.94.pdf and Panopto recording of Lecture 15 on Mar. 22, 2021.

Mar 24: Lecture recording and notes: Math 304-6 Lecture Notes 138.95-147.pdf and Panopto recording of Lecture 16 on Mar. 24, 2021.

Mar 26: Lecture recording and notes: Math 304-6 Lecture Notes 148-155+Webwork.pdf and Panopto recording of Lecture 17 on Mar. 26, 2021.

Mar 29: Lecture recording and notes: Math 304-6 Lecture Notes 156-162.pdf and Panopto recording of Lecture 18 on Mar. 29, 2021.

Mar 31: Lecture recording and notes: Math 304-6 Lecture Notes 163-175.pdf and Panopto recording of Lecture 19 on Mar. 31, 2021.

April 2: Lecture recording and notes: Math 304-6 Lecture Notes 176-188.pdf and Panopto recording of Lecture 20 on April 2, 2021.

April 5: Lecture recording and notes: Math 304-6 Lecture Notes 189-200.pdf and Panopto recording of Lecture 21 on April 5, 2021.

April 7: Lecture recording and notes: Math 304-6 Lecture Notes 201-216.pdf and Panopto recording of Lecture 22 on April 7, 2021.

April 9: Lecture recording and notes: Math 304-6 Lecture Notes 217-218.pdf and Panopto recording of Lecture 23 on April 9, 2021.

April 12: Lecture recording and notes: Math 304-6 Lecture Notes 218.1-218.2.pdf and Panopto recording of Lecture 24 (Review for Exam 2) on April 12, 2021.

April 14: Lecture recording and notes: Math 304-6 Lecture Notes 218.3-218.9.pdf and Panopto recording of Lecture 25 (Review for Exam 2) on April 14, 2021.

April 19: Lecture recording and notes: Math 304-6 Lecture Notes 219-231.pdf and Panopto recording of Lecture 26 on April 19, 2021.

April 21: Lecture recording and notes: Math 304-6 Lecture Notes 232-238.pdf and Panopto recording of Lecture 27 on April 21, 2021.

April 23: Lecture recording and notes: No new notes today, reviewed old notes about eigenvectors, eigenvalues, examples of diagonalization, Exam 2 and webwork problems. Panopto recording of Lecture 28 on April 23, 2021.

April 26: Lecture recording and notes: Math 304-6 Lecture Notes 239-250.pdf and Panopto recording of Lecture 29 on April 26, 2021.

April 28: Lecture recording and notes: Math 304-6 Lecture Notes 251-263.pdf and Panopto recording of Lecture 30 on April 28, 2021.

April 30: Lecture recording and notes: Math 304-6 Lecture Notes 264-276.pdf and Panopto recording of Lecture 31 on April 30, 2021.

May 3: Lecture recording and notes: Math 304-6 Lecture Notes 277-287.pdf and Panopto recording of Lecture 32 on May 3, 2021.

May 5: Lecture recording and notes: Math 304-6 Lecture Notes 288-295.pdf and Panopto recording of Lecture 33 on May 5, 2021.

May 7: Lecture recording and notes: Math 304-6 Lecture Notes 296-298.pdf and Panopto recording of Lecture 34 on May 7, 2021.

May 10: Lecture recording (review for Quiz 10 and Exam 3): Panopto recording of Lecture 35 on May 10, 2021.

May 12: Lecture recording and notes: Math 304-6 Lecture Notes 299-300.pdf and Panopto recording of Lecture 35 on May 12, 2021.

May 17: Lecture recording of review for final exam: Panopto recording of Lecture 36 on May 17, 2021.

Practice exams and actual exams with solutions will be available for download from links posted here.

The following files contain practice exams from previous semesters with material relevant to various Exams. Practice Exam 1, Practice Exam 2, Practice Exam 3, Another Practice Exam 3.

The following file contains a summary of results presented in class and needed for exams. Reading this file is not a replacement for attending class, but could be helpful if you miss some classes because of illness. Here is the link to it: Math304-6 Topics Summary.

The numerial score on each exam will be given a letter grade interpretation, giving each student a letter grade as well as a number grade, and the Total of all points earned will also be given a letter grade interpretation. The letter grades on the exams indicate how a student is doing, and will be taken into consideration in making the interpretation for the Totals. The course grade will be determined by the interpretation of Total points earned. Only borderline cases may be subject to further adjustment based on homework, classroom participation and attendance, as determined by the instructor. Any cases of cheating will be subject to investigation by the Academic Honesty Committee of Harpur College.

Class attendance is required at all scheduled meetings, and sleeping in class does not count as being there. Questions are welcomed at any time during a lecture. At the start of each class be ready to ask questions about homework problems or about the previous lecture. We want to create an atmosphere where you all feel very free to ask questions and make comments. If anyone feels that the instructor has not answered a question clearly, completely, and with respect and consideration for the student who asked it, please let your instructor know about it immediately so he/she can correct the problem. You can do this in class or in office hours, verbally or in writing, on paper or by email, or by whatever means makes you most comfortable, but with enough detail that your instructor can understand what you think was done wrong. It will be too late to help if you only complain at the end of the course. If you are not satisfied by the response of your instructor, please contact the course coordinator, Prof. Alex Feingold.

The material is a combination of theory and calculation, and it is necessary to understand the theory in order to do sensible calculations and interpret them correctly. There is a significant difference between training and education, and we feel strongly that our goal at this university is to educate you, not just to train you to do computations. Theory is not presented to impress you with our knowledge of the subject, but to give you the depth of understanding expected of an adult with a university education in this subject. Some of your instructors have many years of experience teaching mathematics at the university level, but it will require your consistent concentrated study to master this material. While much learning can take place in the classroom, a significant part of it must be done by you outside of class. Using the book, class notes, homework exercises, only you can achieve success in this course. Students who do not take this course seriously, who do not take this advice, are not likely to be rewarded at the end. We are here to help and guide you, and we also grade the exams to judge how much you have learned, but grades are earned by you, not given by us. Exams will be a combination of theory questions and calculations appropriate for a course of this level.

For each section of material covered there will be an assignment of problems from the textbook. Homework will be handled through an online system, and information about how to access it will be provided on the main Math 304 webpage. The homework counts as 5% of your course grade.

Extra topics if time allows (usually not enough time for these in the elementary linear course) :

These paragraphs will be updated before each exam is given.

The list of topics below is most closely related to my lecture notes, but some students have asked to know which sections of the Heffron textbook contain these topics, so I am listing those here: Ch. 1, I. 1-3 (linear systems), III. 1-2 (RREF), Ch. 2, I. 1-2 (vector spaces), II. 1 (linear independence), III. 1-3 (basis, dimension), Ch. 3, I. 1-2 (linear maps), II. 1-2, III. 1-2 (matrix representation of maps), IV. 1-4 (Matrix algebra).

Linear Systems, solving by row reduction of the augmented matrix [A|B] and interpretation in terms of free and dependent variables.

Consistent vs. inconsistent systems. Homogeneous systems AX=O.

Elementary row operations and reduction to Reduced Row Echelon Form (RREF).

Matrices, the set of all mxn real matrices, R^{m}_{n}, addition of
matrices, multiplication of a matrix by a real number (scalar).

The span of a set of vectors in R^{m}_{n} as the set of all linear combinations
from that set.

Matrix shapes, names of special patterns.

Rank of a matrix.

How an mxn matrix A determines a function L_{A}: R^{n}
--> R^{m} by L_{A}(X) = AX.

Linearity properties of the function L_{A}, that is,
L_{A}(X+Y) = L_{A}(X) + L_{A}(Y) for any X, Y in R^{n},
and L_{A}(rX) = r L_{A}(X) for any X in R^{n} and any r in R.

Definition of Ker(L_{A}) and of Range(L_{A}) = Im(L_{A}) and
how to find them by row reduction methods.

Properties of general functions: one-to-one (injective), onto (surjective), both (bijective), invertible.

Connection between properties of matrix A and function L_{A}.

Defintion of matrix multiplication AB through the definition L_{A}
composed with L_{B} equals L_{AB}. Lemma that L_{A}
= L_{B} iff A=B.

Formula for the matrix product of an mxn matrix A with an nxp matrix B giving an mxp matrix
C = AB whose columns are A(Col_{k}(B)) for k = 1, ..., p.

Defintion of standard basis vectors e_{1},
... , e_{n} in R^{n} and lemma that Ae_{j}
= Col_{j}_{}(A), so AX is the sum of x_{j}
Col_{j}(A).

Abstract definition of a real vector space, V. Examples, R^{m}_{n}
is a vector space. For any set S, the set F = {f : S ---> R} of
all functions from S to the reals R, is a vector space.

Definition of a linear transformation L : V ---> W from a vector space to a vector space. Ker(L), Range(L) = Im(L).

Basic facts about vector spaces and about linear transformations (maps), and examples.

Definition and some examples of subspaces.

Definition of when a square matrix is
invertible, uniqueness of the inverse when it exists, and an
algorithm to decide and find it by row reduction of [A | I_{n}].

Definition of transpose of a matrix, of symmetric and anti-symmetric matrices.

Elementary matrices and how they can be used to achieve elementary row or column operations.

The rules of matrix algebra.

The span of a set of vectors S in a vector space V, and why it forms a subspace of V.

How to check that a subset W in V is a subspace of V.

Linear indepdendence or dependence of a subset of V, definition and method of determining that.

Theorems and examples about spanning and independence, connection with rank of a matrix.

Definition of a basis for a vector space, and how to decide if a subset is a basis of V.

Finding a basis for important examples of subspaces, Ker(L), Range(L), where L:V---> W is a linear map.

Dimension of V as the number of vectors in any basis for V.

The standard basis for several examples of vector spaces, including
all the R^{m}_{n} examples and the vector space of polynomials with
degree at most k.

Row-space and Column-space of a matrix, and their dimension related to the rank of the matrix.

Information about the linear transformation
L_{A}: R^{n}--> R^{m} associated with rank(A).

The relationship between the dimensions of Ker(L), Range(L) and V for L:V---> W.

Extending an independent set to a basis, cutting down a spanning set to a basis.

How to represent a general linear map L:V---> W with respect to a choice of basis S in V and basis T in W by a matrix, that is,

using coordinates
with respect to S, [ . ]_{S}, and coordinates with respect to T, [ . ]_{T}, to find
a matrix _{T}[L]_{S}, such that _{T}[L]_{S} [v]_{S} =
[L(v)]_{T}.

The algorithm for finding that matrix by a row reduction of [T | L(S)].

If S and S' are two bases of V, and T and T' are two bases of W, and L:V---> W then there is a
relationship between _{T}[L]_{S}, the matrix representing L from S to T, and
_{T'}[L]_{S'}, the matrix representing L from S' to T'.

That relationship is _{T'}[L]_{S'} =
_{T'}Q_{T} _{T}[L]_{S} _{S}P_{S'}
where _{S}P_{S'} is the transition matrix from S' to S, and
_{T'}Q_{T} is the transition matrix from T to T'.

The concept of isomorphism (bijective linear map) and its properties. Two finite dimensional vector spaces are isomorphic iff they have the same dimension.

Use of a basis S of V to give
coordinates with respect to S for each vector v in V. How that coordinate function, [v]_{S},
is a linear map from V to
R^{n} when a basis S for V consists of n vectors.

Transition matrices which give the relationship between the coordinates of a vector
v with respect to different bases. If S and T are two bases of the same vector space, V,
then the transition matrix from S to T is the square invertible matrix _{T}P_{S} such that
[v]_{T} = _{T}P_{S} [v]_{S}.

Let L:V-->W be a linear map, dim(V) = n, dim(W) = m, S a basis of V and T a basis of W. Then the map
_{T}M_{S} : Lin(V,W)-->F^{m}_{n} defined by _{T}M_{S}(L) =
_{T}[L]_{S} is an isomorphism from the vector space Lin(V,W) = {L:V-->W | L is linear} to
the vector space of all mxn matrices with entries in field F.

Material on determinants, their definition using permutations or by cofactor expansion, their properties, and methods of calculating them (definition by permutations or by cofactor expansions, crosshatching method for matrices of size n = 2 or n = 3 ONLY, using row operations).

The use of determinant to get the characteristic polynomial, det(tI_{n} - A), whose roots
give the eigenvalues of A, and whose expression as a product of powers of distinct linear factors
gives the algebraic multiplicities.

If A and B are similar matrices, so B = P^{-1} A P for an invertible matrix P, then det(B) = det(A) and
the characteristic polynomials of A and B are the same.

Eigenspaces, their properties, and how to decide if a matrix can be diagonalized or not. Theorems about eigenspaces and diagonalizability.

Independence of the union of bases for distinct eigenspaces.

Geometric multiplicity and its relationship to algebraic multiplicity for each eigenvalue,
1 ≤ g_{i} ≤ k_{i}.

Definition and properties of the standard dot product in R^{n}: bilinear, symmetric, positive definite.

Definition of length of a vector, ||v||, for v in R^{n}.

Definition of distance between two vectors, ||u-v||.

Definition of the angle a between two
vectors u and v in R^{n} given by the formula cos(a) = u.v/(||u||)(||v||).

Definition of two vectors in R^{n} being orthogonal (perpendicular) when the angle
between them is a right angle (90 degrees = pi/2 radians), so u.v = 0.

The Cauchy-Schwarz inequality |u.v| <= (||u||)(||v||) and the triangle inequality ||u+v|| <= ||u||+||v||.

Orthogonal and orthonormal subsets, projection of a vector onto another vector,
Proj_{v}(u) = (u.v)/(v.v) v.

A real nxn matrix A is called orthogonal when A^{T} = A^{-1}, its transpose is its inverse.

A real nxn matrix A is orthogonal iff the set of its columns is an orthonormal set in R^{n}.

When A is orthogonal, the linear map L_{A}(X) = AX preserves lengths and angles because
(AX).Y = X.(A^{T}Y) for any X and Y in R^{n}.

The geometrical meaning of projection may be helpful but I would not test it on an exam. Skip decomposing forces.

The projection ṽ = Proj_{W}(v) of vector v into subspace W of R^{n} defined to be
a vector in W such that v - ṽ is orthogonal to W.

If T = {w_{1}, ..., w_{m}} is any basis of W,
then ṽ = ∑_{i=1}^{m} x_{i} w_{i} must satisfy (v - ṽ).w_{j} = 0 for
j = 1,..., m, that is the linear system
∑_{i=1}^{m} x_{i} (w_{i}.w_{j}) = v.w_{j} for j = 1,..., m.

Define the mxm symmetric matrix A = [w_{i}.w_{j}] and the mx1 column matrix B = [v.w_{j}].

This system is just AX = B, and it can be shown that A is invertible, so it always has unique solution
X = A^{-1} B which gives the projection.

When basis T is orthogonal, the matrix A is diagonal and the solution is easy giving an explicit formula
Proj_{W}(v) = ∑_{i=1}^{m} (v.w_{i})/(w_{i}.w_{i}) w_{i}
which is even simpler with T orthonormal.

The Best Approximation Theorem, which says that ṽ = Proj_{W}(v) is the unique
vector in W such that ||v - ṽ|| < ||v - w|| for any w in W distinct from ṽ.

The Gram-Schmidt (G-S) orthogonalization process for converting a basis T = {w_{1}, ..., w_{m}}
of a subspace W to an orthogonal basis T' = {w'_{1}, ..., w'_{m}}.

The formula for each vector in T' is w'_{j} = w_{j} - Proj_{W(j-1)}(w_{j})
for j = 2, ..., m, where W(j-1) is the subspace with orthogonal basis {w'_{1}, ..., w'_{j-1}}
and w'_{1} = w_{1} to start the process.

So from the explicit formula for such a projection, we have Proj_{W(j-1)}(w_{j}) =
∑_{i=1}^{j-1} (w_{j}.w'_{i})/(w'_{i}.w'_{i}) w'_{i}

The Pythagorean Theorem in R^{n} which says that for any orthogonal set
T = {w_{1}, ..., w_{m}}, we have ||w_{1}||^{2} + ... + ||w_{m}||^{2} =
||w_{1} + ... + w_{m}||^{2}.

The standard dot product in the complex vector space C^{n} which is sesquilinear, conjugate symmetric
and positive definite.

If Z and W are column vectors in C^{n} then Z.W = Z^{T} W^{-} where
Z^{T} means the transpose of Z and W^{-} means the complex conjugate of W.

The Hermitian conjugate of a complex nxn matrix A is Ā^{T},
the transpose conjugate of A denoted by A^{H}.

For any column vectors Z and W in C^{n} and any complex nxn matrix A, we have (AZ).W = Z.(A^{H}W).

A complex nxn matrix A is called Hermitian when A^{H} = A.
It is called unitary when A^{H} = A^{-1}.

A is unitary iff the set of its columns is an orthonormal set in C^{n}.

We proved that any real symmetric matrix A = A^{T} in R^{n}_{n} has all its
eigenvalues in the real numbers, and that it is diagonalizable by an orthogonal transition matrix P.

We defined the perp of a subset S in R^{n} to be S^{⊥} =
{X in R^{n} | X.s = 0 for all s in S}.

We proved that S^{⊥} = <S>^{⊥}, the perp of S equals the perp of the
span of S. So to get the perp of a subspace, W, just find the perp of a basis for W.

To find the perp of a subset S = {v_{1},...,v_{m}} in R^{n}, just solve AX = 0
where A is the mxn matrix with Row_{i}(A) = v_{i}^{T} for i = 1,...,m.

We defined the sum of subspaces W = W_{1} + ... + W_{m} in any vector space V.

We showed that sum W is a subspace of V, and defined it to be a direct sum when each vector w in W has
a unique expression as w = w_{1} + ... + w_{m} for w_{i} in W_{i}.

We discussed basic examples of sums and direct sums in R^{2} and R^{3}.

We discussed the theorem which says that a sum W (as above) is direct iff the intersection of W_{i}
with the sum of all other W_{j} is only the zero vector, for each i = 1,...,m.

We discussed the theorem that if B_{i} is a basis of W_{i}, then the sum W is direct iff
the long list B of all the vectors in B_{1} ... B_{m} is a basis of W.

We defined a direct sum of subspaces in R^{n} to be an orthogonal direct sum when those subspaces
are mutually orthogonal, that is, W_{i}⊥W_{j} for i≠j.

We proved the theorem that for any subspace W in R^{n}, that W + W^{⊥} = R^{n}.

We applied these ideas to the eigenspaces of a real symmetric nxn matrix A = A^{T}, and showed that
R^{n} is the orthogonal direct sum of all those eigenspaces.

We then got that any real symmetric nxn matrix A is orthogonally diagonalizable, D = P^{T}AP,
for an orthogonal transition matrix P.

Anything from any part of the course.

This page last modified on 12-2-2022.