Those have no knowledge of matrix operations:
It will be helpful if you watch the following nine(9) videos on Matrices at Khan Academy. They are quite easy.
Once you digest these nine videos, you can watch the other videos on Matrices at Khan Academy
Though I will introduce all these basic knowledge during classes, the speed of the lecture may appear very fast. It will make your Math 332 easier if you know these basic matrix operations. If you need help, you can come by during my office hours.
Videos:
Multiplying matrices by matrices <-Important !
In each file below, you will first have a full lecture note, followed by a 'incomplete' note, some of which I distributed as handouts to use during lecture.
Review of linear system, Augmented matrix, Elementary row operations, Geometry in linear system
1.2 Gaussian-Jordan Elimination, General solution, half way. I will continue next week.
Row echelon form, Reduced row echelon form, G-J Elimination method, Example General Solution via Augmented matrix and G-J methods,
Reduced echelon form and the Solution types, Echelon form and the Solution types
Recommended/In-class exercises ; No need to submit but I recommend you try them
In-class exercises in the lecture files.
HW #1 for submission Due 1/20 Tuesday 3:15pm at the beginning of class.
Only a paper submission is accepted. Your solution must be handwritten. See Course Syllabus for detail.
'Exercises 1.1' below means, Exercises at the end of Section 1.1 in our textbook.
Exercises 1.1: #12, 20b, TF(e)(f)(g).
Instruction:
When solving a linear system, you can (i) solve the linear equations directly or (ii) (I strong recommend that) you set up the augmented matrix and then apply elementary row operations, that is, Gauss-Jordan elimination process to rows.
For T/F questions, "justify your answer" means that
If true, then state the reason (i.e., prove) why it's true;
If false, give a counterexample.
1.2 Gaussian-Jordan Elimination, General solution
Row echelon form, Reduced row echelon form, G-J Elimination method, Example General Solution via Augmented matrix and G-J methods,
Reduced echelon form and the Solution types, Echelon form and the Solution types
Homogeneous Linear Systems are always consistent, Geometry in Homogenous Lin System
1.3 Matrix Terminologies and Operations, half way. I will continue next week
Recommended/In-class exercises ; No need to submit but I recommend you try them
In-class exercises in the lecture files.
HW #2 for submission Due 1/27 Tuesday 3:15pm at the beginning of class.
Only a paper submission is accepted. Your solution must be handwritten. See Course Syllabus for detail.
Instruction:
When solving a linear system, set up the augmented matrix and then apply elementary row operations, that is, Gauss-Jordan elimination process to rows. <-- required!
Specify an elementary row operation applied at each step while performing G-J process.<-- required!
For T/F questions, "justify your answer" means that
If true, then state the reason (i.e., prove) why it's true;
If false, give a counterexample.
Exercises 1.1: #16b (Use Elem row op to augmented matrix. For its general solution, use parameter(s) or all the raw variables as done in class.)
Exercises 1.2: #8 and 12 (solve 8 and 12 in continuation/ together), 24a,c, 26 (The case analysis for the values of a must be done clearly), 31, 43a,b, 38 (Give a general solution for a,b,c, and d. Warning: a being not equal to 0 is not guaranteed hence you can't divide the eqn by a in the first place), 39 (Argue via G-J and ref and rref. Not allowed to use advanced materials, such as invertibility or determinants)
Exercises 1.2: #40 (For item b, it is not required but I recommend you use "free variables" instead of "parameters"), TF(b)(d)(g)(i)
Exercises 1.3: #5d,e,h,i, 16, 32(a)(c), 36b, TF(m) postponed to the next HW set
1.3 Matrix Terminologies and Operations
1.3 Matrix Multiplication: Columns and Rows
Discussing #1 in my Exam 1 Fall 2024
1.3 Matrix Equation, Linear System, and Linear Combination (Def)
1.4 Matrix Algebra: Properties/Non-properties, Zero matrix, Identity matrix
1.4 Matrix Algebra: Inverse, Powers, Transpose, started. Will continue next week.
Recommended/In-class exercises ; No need to submit but I recommend you try them
In-class exercises in the lecture files.
HW #3 for submission Due 2/3 Tuesday 3:15pm at the beginning of class.
Only a paper submission is accepted. Your solution must be handwritten. See Course Syllabus for detail.
Exercises 1.3: #5d,e,h,i, 16, 32(a)(c), 36b, TF(d)(m)
Exercises 1.3: #24 (set up a (new) auxiliary linear system from the given matrix equality, perform the row operations to the augmented matrix of the system, then compute the final values of a, b, c, d)
1.3 Matrix Multiplication: Columns and Rows: "1.3 Class exercise: Rows and Columns of AB" on the last page
1.3 Matrix Equation, Linear System, and Linear Combination (Def): "1.3 Class exercise: Ax in Linear Combination of Column vectors" on the last page
Exercises 1.4: #36, TF(j)
For #36 and TF(j): Explain through matrix multiplications via rows/columns in 1.3 Matrix Multiplication: Columns and Rows. Do NOT use any advanced theories that appear in a later section such as the relation of a row reduced echelon form and invertibility if you know what I mean.)
1.4 Matrix Algebra: Inverse, Powers, Transpose
Discussing #6a in my Exam 1 Fall 2024
1.5 Elementary matrices, Inverse Algorithm, Invertible matrix TFAE (a)-(d)
Algorithm for Invertibility. Elementary matrices, intro and being invertible, Usefulness of elementary matrices, Invertible Matrix and its equivalent statements (a)--(d) including a homogeneous system.
Discussing #5b in my Exam 1 Fall 2024
Recommended/In-class exercises ; No need to submit but I recommend you try them
In-class exercises in the lecture files.
HW #4 for submission Due 2/10 Tuesday 3:15pm at the beginning of class.
Only a paper submission is accepted. Your solution must be handwritten. See Course Syllabus for detail.
Exercises 1.4: #31(b)(c), 33(a), 43, TF(k)
Exercises 1.4: #40, 50
Exercises 1.5: #6b, 8c, 16, 22, 32, TF(d)
1.6 Non-homogeneous Ax=b, Invertible matrix TFAE (e)(f)
1.7 Diagonal, Triangular, and Symmetric Matrices
2.1 Determinant by Cofactor expansion and Determinants of Three special matrices
2.2 Determinants by Row or Column reduction
2.2-2.3 Property or Non-property, covered most of them. Will finish next week
Recommended/In-class exercises ; No need to submit but I recommend you try them
In-class exercises in the lecture files.
HW #5 for submission Due 2/17 Tuesday 3:15pm at the beginning of class. at the end of class; see remark #2 below .
Only a paper submission is accepted. Your solution must be handwritten. See Course Syllabus for detail.
Remarks:
This homework set is long. So, I convert some of the questions to "optional", which means they will not be graded but the solutions will be provided. Those problems have been labelled "optional" and been greyed out below.
The questions 2.2.20, 2.2.26, 2.3.36 are marked by *. We couldn't cover all the materials for you to do these three questions. During 2/17 class, we will work them out together and you include them into your HW #5 submission. (I really want you to practice those questions with care as they will be on the Midterm 1)
Exercises 1.6: #14(see Example 3 on pp.8-9 in 1.6 Non-homogeneous Ax=b, Invertible matrix TFAE (e)(f).
Do (a)--(d) as in the lecture file, that is
(a) Rewrite the given linear system in a matrix equation Ax=b
(b) Find the conditions on the b_i's for the system to be consistent, as asked in the exercise
(c) Integrate the conditions on the b_i's found in (b) to rewrite the solution b in column vector in independent variable(s); be sure to give the range of the independent variable(s)
(d) What can you say about the invertibility of A?
Exercises 1.6: #20 (optional), 22 (optional), T/F(f).
Exercises 1.7: #14, 33 (Do only the part "verify 1.7.1(b)", no need to do 1.7.1(d)), 18, 20 (Hint: "by inspection" indicates you can quickly recognize the invertibility because it is a triangular matrix)
Below, when evaluating determinants, use row or column operations; do NOT use a cofactor expansion or a short cut for a 3by3 matrix.
Exercises 2.2: #20*, 24 (optional; not via Cofactor expansion but use row reductions), 26*, 34 (clearly specify what row/column operations you apply)
Exercises 2.3: #34, 36*, TF(c)
Chapter 2 Supplementary Exercises: #33 (Extra Credit problem; You are not required to do this question but if you do, you will earn extra credits)
Make sure to read Remarks above.
2.2-2.3 Property or Non-property, finished
2.3 Determinant test for Invertibility. RREF and Invertibility. Invt TFAE
4.1 Vector Spaces, about a quarter of the file done.
Definition of vector space
Examples and non-examples of Vector Spaces
No homework due 2/24. Next homework, HW 6, will be due 3/3.
Definition of vector space
Examples and non-examples of Vector Spaces
HW #6 for submission Due 3/3 Tue. 3:15pm at the beginning of class.
Only a paper submission is accepted. Your solution must be handwritten. See Course Syllabus for detail.
Exercises 4.1: #4,7,8,9.
In Section 4.1, to determine a vector space:
You must show Steps (1)--(4) (1&2 for setting and 3&4 for 10 axioms), as done in class, then state your conclusion clearly, if showing 'a vector space'. If one of the axioms violates, you don't have to go through everything; you can just show an appropriate counterexamples that violates the axiom.
In Step(1), you must elaborate the set V using the set notation {....| ...........} <--- Very important!
Sometime later in future, you may omit the detailed checking of Axiom 2-5 and 7-10 when addition + and scalar multiplication * are standard operations; but, not yet!
Definition of Subspace and "Subspace test"
Examples and non-examples of subspaces
HW #7 for submission Due 3/10 Tue. 3:15pm at the beginning of class.
Only a paper submission is accepted. Your solution must be handwritten. See Course Syllabus for detail. See my email for an alternative method
Exercises 4.2: #1(c) and 2(a) (<=do these questions together). #3(b), #4(d), #18, #24(a), #TF(e)
In Section 4.2, to use "Subspace Test": You must show (0) & (1)--(3) (note that (3) consist of two "closures" to check), as done in class, then state your conclusion clearly if showing 'a subspace'. If one of the axioms violates, you don't have to go through everything; you can just show an appropriate counterexamples that violates the axiom.
4.3 (&1.3) Linear combinations & Task0
4.3 (& 1.3) Lecture video 1 : Linear combination
4.3 (& 1.3) Lecture video 2 :Typical task #0; Determining whether a vector is a linear combination of a set of vectors v_1, v_2,....,v_n
4.3 no1 Lecture video 1: Span, Intro
4.3 no1 Lecture video 2: Typical task #1; Determining whether a vector is in Span(S)
4.3 4.4 4.5 4.6 4.8 Table Examples - guide
4.3 4.4 4.5 4.6 4.8 Explanatory video: Fill out the answers, as we do examples in class. It helps that you are organized and clear on what we are doing!
Here's its filled note 4.3 4.4 4.5 4.6 4.8 Table Examples - full
4.3 no2 Span: Subspace, how large? Task2:
4.3 no2 Lecture video 1: Span(S) is a subspace
4.3 no2 Lecture video 2: Typical task #2; Determine how large can Span(S) be in V, that is, Span(S)=V or not
4.3 no2 Lecture video 3: Typical task #2; (Span(S)=V or not) and Invertibility (to be aware but not allowed use in 4.5 & 4.4)
4.3 no3 Swapping Spanning Sets, Task3
4.3 Lecture video: When can two spanning sets be swapped? to be uploaded, perhaps..... Will do after Spring break
HW #8 for submission Due 3/24 Tue. 3/26 Thur. 3:15pm at the beginning of class. Please check my emails for why the deadline was postponed & what you have to fix in your solutions regarding the polynomials.
Only a paper submission is accepted. Your solution must be handwritten. See Course Syllabus for detail.
Recommend that you follow the steps as done in class & lecture files
To figure
Vector equation (in terms of the raw vectors given in the questions, not in Euclidean vectors)
(See if you can do eyeballing to find the coefficients in the vector equation above. If you can, complete the vector equation with those coefficients; otherwise, proceed to Matrix eqn & Solving.. Either way, you will have to clearly state "Reasoning...." and "Conclusion" at the end.)
Matrix equation
Solving the Matrix eqn via row operations
Reasoning, such as interpreting your solution in terms of the given question
Conclusion
Required to give in the steps above for all the problems except #TF(c).
Vector Equation (in terms of the raw vectors given in the questions, not in Euclidean vectors)
Reasoning, such as interpreting your solution in terms of the given question
Conclusion
Exercises 4.3: #1(a)(b)(c), #3(a), #4(a)
In addition, If a linear combination, display the linear combination with appropriate coefficients, as done in class
Exercises 4.3: #8(b)(c)
In addition to determining whether the given vector is contained Span S (as in the exercises);
if the given vector is contained in Span S, then express the given vector as a linear combination of the vectors of S, as done in class.
Exercises 4.3: #7(b), #10
In addition to determining whether Span S=V or not (as in the exercise);
if Span S is not equal to V, then express Span S using a set notation, as done in class.
Exercises 4.3: #TF(c)
4.3 no3 Swapping Spanning Sets, Task3
4.4 Linear Independece / Dependence, Basics
Definition
Basic process to determine Lin. Indep/Dep.
4.4 4.6 Linear Independece / Dependence, Theorems
Geometric meaning of Lin. Indep/Dep.
Lin indep/dep of a set of vectors under various scenarios:
(i) the set includes the zero vector, (ii) the set consists of two vectors, (iii) when V=R^n and the set contains strictly more vectors than n.
S, subset of S, superset of S, and their Spanning subspaces
Plus/Minus Theorem
HW #9 for submission Due 3/31 Tue. 3:15pm at the beginning of class.
Exercises 4.3: #19
Exercises 4.4 #2(a), #4(a), #5(a); if linearly dependent, display a linear dependency of the vectors, as done in class.
"Additional Instructions for HW8" above still applies.
Exercises 4.4 #22, #TF(d)
Exercises 4.4#26, #29 postponed to next week
Additional HW for 4.3-4.4 #1 and #2 <-- Click !!!!!!!!
4.5-4.6-4.7 Basis Dimension Coordinates
4.5-4.6 Basis and Dimension Introduction
Definition of Basis and Dimension
Tasks: (a) Determine if the given set S of the finite vectors of V is a basis for V; (b) Find dim(V) in the handout 4.5-4.6-4.7 Basis Dimension Coordinates
Note of two different methods : Method 1 (full length) Basic methods via Definitions; Method 2 (short cut) Invertibility of A;
for #1 (4.5 Examples 5&9 Euclidean vector space) - The matrix A is immediate to form.
for #2 (4.5 Exercise 17 Polynomial space) - You first need to produce a linear system Ax=b and/or Ax=0 as in Method 1, then apply the invertibility of A as in Method 2.
4.6 Dimension, Minimal Spanning set, Maximal Independent set
Dimension and Min/Max of Spanning/Indep Set (Thm 4.6.2)
Tasks: (a) Determine if the given set S of the finite vectors of V is a basis for V; (b) Find dim(V) in the handout 4.5-4.6-4.7 Basis Dimension Coordinates
Note of the third method: Method 3 (via Dim and Min/Max of Spanning/Indep set & mix of Methods 1 & 2)
4.7 & 4.5 Ordered Basis, Coordinates, and Change of Bases (draft; the current version has all the contents that were covered till 4/2, but it will be updated when I have time)
Ordered Basis, Coordinate vectors relative to the ordered basis
Change of basis, Transition matrix, Relation between the two transition matrices
HW #10 for submission Due 4/7 Tue. 3:15pm at the beginning of class.
Exercises 4.4#26, #29
Exercises 4.5 #4,#6;
Additional Instructions:
State the dimensions as well, as done (a) & (b) in the examples #1 & #2 on p.6 & p.9-11 in 4.5-4.6 Basis and Dimension Introduction )
These HW exercises are NOT on the Euclidean space. If you use the short cut (as Method 2 in #1), your solution must include the procedure leading to a linear system (or a matrix equation) to have a matrix A. See the descriptions under Lecture above. Also, refer to the example #2 on p.10-11 in 4.5-4.6 Basis and Dimension Introduction
Exercises 4.5 TF (a), (b), (c)
Comments:
All 4.5 TF questions are based on the definition of a basis.
On the homework, each question is worth 2 pts; 1 pt for a correct T/F and 1 pt for a justification. If similar T/F questions are asked on the exams, you will be asked to answer only T/F without a justification.
Exercises 4.6 TF(b), (c), (d), (e), (i)
Comment:
All 4.6 TF questions are based on the 'optimal status of a basis' between Min-of-Span and Max-of-Indep on pp.2-3 in 4.6 Dimension, Minimal Spanning set, Maximal Independent set
On the homework, each question is worth 2 pts; 1 pt for a correct T/F and 1 pt for a justification. If similar T/F questions are asked on the exams, you will be asked to answer only T/F without a justification.
Extra Credit: Exercises 4.6 #14
Comment:
Grading for extra credit questions will be strict. Your argument must be logical and cohesive - writing random facts here and there is not a proof.
Additional HW for 4.7 & 4.5: #1c,d in the handout 4.5-4.6-4.7 Basis Dimension Coordinates
Exercises 4.5 #25 (For (b), interpret the basis as an ordered basis.)
4.7 & 4.5 Ordered Basis, Coordinates, and Change of Bases (draft; the current version has all the contents that were covered 4/2, but it will be updated when I have time)
Ordered Basis, Coordinate vectors relative to the ordered basis <-- done last week
Change of basis, Transition matrix, Relation between the two transition matrices
We covered theoretical aspect of this topic during the 4/7 class. It will be asked only as extra credit question on the final exam. (I elaborated the reason during class.) I will assign a homework question as an 'optional' question.
The detailed theory discussed on 4/7 is not on the lecture file.
You are welcome to come by during my office hours (or set up an appt with me) if you would like to learn more about the topic.
4.7-4.8 Given a matrix A, Col(A), Row(A), rank(A), Null(A), nullity(A)
In order to understand the topic, it is imperative that you did Additional HW for 4.3-4.4 #1 and #2 from HW 9. Though my lecture file includes the contents of the homework, it will be easier for you to go through the homework to see the point.
Midterm 2
5.1 Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors of a matrix, Characteristic polynomial and characteristic equation of a matrix; Eigenspace of a matrix w.r.t. an eigenvalue, Finding bases for the eigenspaces of a matrix <-- No lecture file available
5.2 Diagonalization
Characterization of diagonalizable matrices in terms of eigenvectors. How to check whether or not a matrix is diagonalizable; The dimensions of each Eigenspace should sum up exactly n. <-- No lecture file available
Procedure for diagonalizing a matrix, relation between P and D in the diagonalization, <-- No lecture file available
Convenience of diagonalization (if possible); e.g. Powers of matrices <-- No lecture file available Postponed to the next HW set... perhaps
HW #11 for submission Due 4/21 Tue. 3:15pm at the beginning of class.
Exercises 4.7 #12 (optional)
This question focuses on the conceptual meaning of a transition matrix in changing bases, rather than just the formula.
This question will not be graded and will not be part of HW11 score. However, the solution will be provided later.
Will help solving a theoretical problem on the change of bases that will appear as an extra credit problem on the final exam, as we discussed in class
Exercises 4.8 #16
In addition, state the dimension of the subspace spanned by the given vectors after finding its basis.
Exercises 4.8 #7(b):
In addition, state the dimension of the solution space of Ax=0.
Express the general solutions and your basis using the SET notations, as done in class.
For a "vector form", see p.13 in 4.7-4.8 Given a matrix A, Col(A), Row(A), rank(A), Null(A), nullity(A)
Exercises 4.8 #10(a): <-- I originally posted #10, but as of 4/15, it requires only item (a) for #10.
In addition, state the dimension of Null(A) and Row(A).
Always, provide a basis clearly using the set notation, as done in class.
Exercises 4.8 #18 (Read Example 6 first to understand what the question is asking if you aren't sure. )
Exercises 4.9 #6, #8
Exercises 4.9 #24(a), #31,
Exercises 4.9 One of #40 or #TF(b) (Observe the connection between the two statements).
Hint: Consider A as m x n matrix and look at linear independence of rows and columns when m less than n and when m larger than n.
Exercises 5.1 #8, #34
Exercises 5.2 #8, #20(a), #One of TF(d) or (e) Postponed to the next HW set... perhaps
My aim is to cover the following sections. However, we may not have enough to time to do all, and I will have to adjust what to cover/what to omit, or the order of the topics, as the classes progress.
5.2 Diagonalization
Convenience of diagonalization (if possible); e.g. Powers of matrices <-- No lecture file available; Briefly covered. Will not be asked on the exam.
6.1 Examples of Inner Product Space - usual & unusual
Different Inner products on R^n (Standard Euclidean inner product vs. Weighted Euclidean inner product), Inner products on Matrices, Polynomials (Standard polynomial inner product vs. Integral polynomial inner product), and Continuous functions
Induced Norm and distance functions
6.1 Norm and distance functions and their properties
6.2 Inequalities and Orthogonality
pp.7-onward only covered this week.
pp.1-6 Various inequalities are postpone to next week .
6.2 Orthogonal complement May be covered next week if time permits.
6.3 Orthogonal / Orthonormal Sets and Bases Started.
6.3 Orthogonal / Orthonormal Sets and Bases - handout
We may not have enough to time to do all, and I will have to adjust what to cover/what to omit, or the order of the topics, as the classes progress.
Suggested HW, not required to submit: Fill out Handout 6.3 Orthogonal / Orthonormal Sets and Bases - handout as instructed in class to be better prepared for the next class.
HW #12 for submission Due 4/28 Tue. 3:15pm at the beginning of class.
Exercises 6.1 #(Two of 18, 20, 22), #28, #34, #38
Exercises 6.2 #17,
Exercises 6.2 #27; Do three parts: (a), (b),(c) as below.
(a) Basis
You must show the process as done in class; refer to 6.2 Orthogonal complement
Initial setting: the system of the inner product between each given vector and a vector of the orthogonal complement equal to zero,
Linear system in matrix form Ax=0
Solving the matrix system by augmented matrix and row operations
Express W^{ortho} = Span of {spanning vectors} and write a basis B={....} clearly.
(b) According to your setup, specify if the subspace W spanned by v_1, v_2, v_3 is (i) Col(A), (ii) Row(A), (iii) Null(A) (choose one correct answer)
(c) According to your setup, specify if the orthogonal complement W^{ortho} that you found is (i) Col(A), (ii) Row(A), (iii) Null(A) (choose one correct answer)
Exercises 6.2 #37
Exercises 6.3 #3b, #4a, #5, #10
6.3 Orthogonal / Orthonormal Sets and Bases - handout
My aim is to cover the following sections. However, we may not have enough to time to do all, and I will have to adjust what to cover/what to omit, or the order of the topics, as the classes progress.
6.3 Orthogonal / Orthonormal Sets and Bases
Comment on Theorem 6.3.2 (from class)
The 'orthonormal' version (b) is simple and worth to memorize while the 'orthogonal' version (b) is too complicated to memorize. I do not recommend you blindly memorize (b); you will forget (b) almost immediately. Alternatively, I recommend the following approach:
(i) normalize the given vectors to obtain an orthonormal basis
(ii) perform the inner product between u and each v_i
Besides, what's really meaningful is to understand the component of u in the direction of v_i for each i
When applying 6.3.2 to compute with actual numbers, I basically repeat the proof process, which is much easier than trying to apply (a).
6.3 Gram-Schmidt Process to construct an orthogonal set of in an inner product space <-- Will NOT be asked on the final as regular problems. A theoretical question will be asked as an extra credit problem. Spend time on this topic only after you master all other regular topics.
6.2 Inequalities and Orthogonality
pp.7-onward only covered this week.; Done last week
pp.1-6 Various inequalities
Will be updated further after the 4/30 class.
We may not have enough to time to do all, and I will have to adjust what to cover/what to omit, or the order of the topics, as the classes progress.
HW #13 NOT to be submitted. I will specify what problems to work on to be prepared for the final exam. Will be updated further after the 4/30 class.
Exercises 6.2 #27; Do three parts: (a), (b),(c) as below.
(a) Basis
You must show the process as done in class; refer to 6.2 Orthogonal complement
Initial setting: the system of the inner product between each given vector and a vector of the orthogonal complement equal to zero,
Linear system in matrix form Ax=0
Solving the matrix system by augmented matrix and row operations
Express W^{ortho} = Span of {spanning vectors} and write a basis B={....} clearly.
(b) According to your setup, specify if the subspace W spanned by v_1, v_2, v_3 is (i) Col(A), (ii) Row(A), (iii) Null(A) (choose one correct answer)
(c) According to your setup, specify if the orthogonal complement W^{ortho} that you found is (i) Col(A), (ii) Row(A), (iii) Null(A) (choose one correct answer)
Exercises 6.3 #3b, #4a, (submitted in HW12), #5, #10 (See Comment on Theorem 6.3.2 (from class) above under Lecture.)
Exercises 6.3 (Gram-Schmidt process) #29, #36, #37 <-- Will NOT be asked on the final as regular problems. A theoretical question will be asked as an extra credit problem. Spend time on this topic only after you master all other regular topics.