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. 3:15pm at the beginning of class.
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 coeffients; 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)
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)