| 1: Vector & Linear Equations |
Live #1 |
Youtube #1 |
Problems #1 |
Notes #1 |
2 Equations & 2 Variables: We really begin by examining a system of equations with 2 equations and 2 variables. We examine the system with (1) The Row Picture and (2) The Column Picture. |
Solution Set 1.1 |
| 2: Vector & Linear Equations II |
Live #2 |
Youtube #2 |
Problems #2 |
Notes #2 |
3 Equations & 3 Variables: We advance to a system of 3 equations with 3 variables. Once again, we examine the Row Picture and the Column Picture |
Solution Set 1.2 |
| 3: Vector & Linear Equations III |
Live #3 |
Youtube #3 |
Problems #3 |
Notes #3 |
Matrix Picture: I stay with the 3 variable, 3 equation system, but now I introduce the Matrix Picture, which I use to verify the solution to a system here. In the next lecture, we go deeper into Matrix Notation, including Ax=b form, identifying entries in a matrix using A(i,j) notation, and understanding Matrix-Vector multiplication through this new notation. |
Solution Set 1.3 |
| 4: Vector & Linear Equations IV |
Live #4 |
Youtube #4 |
Problems #4 |
Notes #4 |
Fundamental Question & Matrix Notation: I look at the 3 pictures: (1) Row Picture, (2) Column Picture, (3) Matrix Picture -- first for a specific 3-D system, and then for a general R^n system. I connect the Matrix Picture to the System to (one of) the Fundaemental Questions of Linear Algebra: How to solve systems of Linear Equations, and I explain how that question is the same as How to solve Ax=b. |
Solution Set 2.1 |
| 5: Vector & Linear Equations V |
Live #5 |
Youtube #5 |
Problems #5 |
Notes #5 |
Matrix Notation II: A deeper explanation of Matrix Notation -- I discuss identifying entries, provide a specific example of Matrix-Column Multiplication, and generalize this using Matrix Notation. |
Solution Set 2.2 |
| 6: Vector & Linear Equations VI |
Live #6 |
Youtube #6 |
Problems #6 |
Notes #6 |
The Outliers: Now we look at the fringe cases: 3 equations and 3 variables but no solutions? I look at the Row Picture (not ready for Column Picture yet), and demonstrate the four possible solution spaces for three planes intersecting. |
Solution Set 2.3 |
| 7: Vectors & Linear Equations VII |
Live #7 |
Youtube #7 |
Problems #7 |
Notes #7 |
Vectors: We've talked a lot about Linear Equations; Now onto Vectors. I cover Vector Addition (Analytically + Parallelogram Method) and Vector Subtraction (Analytically + Graphically Switch Direction), and show that Vector Addition & Subtraction are really part of one parallelogram defined by the two vectors which we're adding/subtracting. |
Solution Set 2.4 |
| 8: Vectors & Linear Equations VIII |
Live #8 |
Youtube #8 |
Problems #8 |
Notes #8 |
Linear Combinations: Now I formally introduce the concept of Linear Combinations, having already demonstrated Vector Addition and Vector Subtraction and Scalar Multiplication. I begin by specific example for all four of these ideas and then generalize all of them. |
Solution Set 2.5 |
| 9: Vectors & Linear Equations IX |
Live #9 |
Youtube #9 |
Problems #9 |
Notes #9 |
Span: I come to the idea of span! I informally introduce all possible linear combinations of vectors: this can be nothing (!), a line, a plane, or even all of 3D space! Or it could be some hyperplanes if we go further, but I stay on the ground for now. I plan to further elaborate on these ideas on the next few lectures. |
Solution Set 2.6 |
| 10: Vectors & Linear Equations X |
Live #10 |
Youtube #10 |
Problems #10 |
Notes #10 |
3D Span: I take one more step this time: from all linear combinations of two 2D vectors being the entire R^2 plane to all linear combinations of three 3D begin the entire R^3 3D space. I begin with a specific linear combination of three vectors. Then, I go on to generalize the constants c, d, and e. I then generalize the components of the three vectors. I welcome the idea of take the two vectors that span the plane, and then adding a third vector that enables us to span *all* the planes in R^3. I end with an animation and an interactive demonstrating Span in both 2D and 3D. Check out the great 2D R^2 Span, 3D R^3 Span (by planes), and 3D R^3 Span (all of Space) |
Solution Set 2.7 |
| 11: Vectors & Linear Equations XI |
Live #11 |
Youtube #11 |
Problems #11 |
Notes #11 |
Span (Formal): I finally introduce the concept of Span, using formal notation! I also use review what all the linear combinations of 1 Vector and 2 Vectors (all distinct) in R^2 look like, and 3 Vectors (all distinct) in R^3 look like using the new notation. I further use some animations to demonstrate the content of the past few lectures. |
| 12: Vectors & Linear Equations XII |
Live #12 |
Youtube #12 |
Problems #12 |
Notes #12 |
Special Cases: Now I come to the failures of span: What happens when the Span of a Vector isn't just straight R^1? I stay with the Span of just one Vector u here, to build the idea slowly. In the next lecture, I hope to steadily introduce the idea that the Span of 2 two-dimensional vectors need not be R^2, but can be R^1, or even R^0. Adventure awaits! |
| 13: Vectors & Linear Equations XIII |
Live #13 |
Youtube #13 |
Problems #13 |
Notes #13 |
Special Cases II: We saw before that Span(u)=R^1 or R^0. Now we see that Span(u,v)=R^2,R^1, or R^0. I begin the lecture by asking what Span(u,v) *usually* is, but then demonstrate the other two scenarios, making sure to use (1) Notation, (2) Translation, (3) Geoemtric Picture for each of the three scenarios. I end by hinting what the next lecture may be about. |
| 14: Vectors & Linear Equations XIV |
Live #14 |
Youtube #14 |
Problems #14 |
Notes #14 |
Special Cases III: You might start seeing the pattern by now. But here, we see that Span(u,v,w)=R^3,R^2,R^1, or R^0. Once again, I examine each of the four scenarios through (1) Notation, (2) Algebra, (3) Translation, (4) Geometric Picture. |
| 15: Vectors & Linear Equations XV |
Live #15 |
Youtube #15 |
Problems #15 |
Notes #15 |
Dot Products: This is literally the bones and heart of applied mathematics! We come to the idea of Dot Products. We already learned Vector Addition and Subtraction, but we totally avoided Vector Multiplication. Here, I define the dot product in its two definitions and demonstrate its utility with both defintions. |
| 16: Vectors & Linear Equations XVI |
Live #16 |
Youtube #16 |
Problems #16 |
Notes #16 |
Dot Product Properties: Dot Product Properties: I begin examining the utility of the dot product with three properties: (1) Magnitude of a Vector (2) Dot Product of a Vector with itself (3) The Commutativity of the Dot Product |
| 17: Vectors & Linear Equations XVII |
Live #17 |
Youtube #17 |
Problems #17 |
Notes #17 |
Dot Product Properties II: Today I examine solely one property of the dot product: (a+b) 'dot' (a+b) |
| 18: Vectors & Linear Equations XVIII |
Live #18 |
Youtube #18 |
Problems #18 |
Notes #18 |
Dot Product Properties III: Investigating the difference of two vectors via the dot product. |
| 19: Vectors & Linear Equations XIX |
Live #19 |
Youtube #19 |
Problems #19 |
Notes #19 |
Dot Product Properties IV: How can a dot product help us understand the orthogonality of two vectors? This is the pivotal question I seek to answer in this lecture. |
| 20: Vectors & Linear Equations XIX |
Live #20 |
Youtube #20 |
Problems #20 |
Notes #20 |
Dot Product Properties IV: Now we examine an alternate definition of the dot product -- this being the Sigma Notation Definition -- more portable, but equally harder to understand. |
| 21: Vectors & Linear Equations XIX |
Live #21 |
Youtube #21 |
Problems #21 |
Notes #21 |
Unit Vectors: I introduce the idea of (1) The Unit Vector (2) Finding the direction of a vector |
| 22: Vectors & Linear Equations XIX |
Live #22 |
Youtube #22 |
Problems #22 |
Notes #22 |
Cauchy Scwarz Inequality: I review the Cauchy Scwarz Inequality and prove the Triangle Inequality using the Cauchy Schwarz Inequality |
| 23: Vectors & Linear Equations XIX |
Live #23 |
Youtube #23 |
Problems #23 |
Notes #23 |
TBD: TBD |
| 24: Vectors & Linear Equations XXIV |
Live #24 |
Youtube #24 |
Problems #24 |
Notes #24 |
AM-GM: I establish the Arithmetic Mean - Geometric Mean Inequality (AM-GM) via the Cauchy Schwarz Inequality and via the binomial theorem. What could be better! |
| 25 : Vectors and Linear Equations XXV |
Live #25 |
Youtube #25 |
Problems #25 |
Notes #25 |
Rotation Problem: Occasionally I examine very interesting problems that try to relate much of what we've learned thus far. Today, I examine how we can find the result of rotating a point across the origin. |
| 26 : Vectors and Linear Equations XXVI |
Live #26 |
Youtube #26 |
Problems #26 |
Notes #26 |
Angle between Lines: Occasionally I examine very interesting problems that try to relate much of what we've learned thus far. Today, I examine how we can find the angle between two lines using the second definition of the dot product. |
| 27 : Vectors and Linear Equations XXVII |
Live #27 |
Youtube #27 |
Problems #27 |
Notes #27 |
Simpler Acute Angle: I look at the same problem again, but much simpler this time! Although I do believe that is a relative word. |
| 28 : Vectors and Linear Equations XXVIII |
Live #28 |
Youtube #28 |
Problems #28 |
Notes #28 |
Possible Solution Sets: Now we examine the various possible solution sets of linear equations. This is a very important lesson, mind you. |
| 29 : Vectors and Linear Equations XXIX |
Live #29 |
Youtube #29 |
Problems #29 |
Notes #29 |
Matrix Notation: Today we look at Matrix Notation and Matrix-Vector Multiplication; We also set up for generalizing matrix vector multiplication and examining it via the dot product, which we'll do tommorow. |
| 30 : Vectors and Linear Equations XXX |
Live #30 |
Youtube #30 |
Problems #30 |
Notes #30 |
A Second View of Matrices: Taking another look at matrix vector multiplication
|
| 31: Vectors & Linear Equations XXXI |
Live #31 |
Youtube #31 |
Problems #31 |
Notes #31 |
Matrix Elimination: Now we begin Matrix Elimination. Finally! |
| 32: Vectors & Linear Equations XIX |
Live #32 |
Youtube #32 |
Problems #32 |
Notes #32 |
Solving Systems with Elimination: Now I solve systems of linear equations with elimination |
| 33: Vectors & Linear Equations XIX |
Live #33 |
Youtube #32 |
Problems #33 |
Notes #32 |
Solving Systems with Elimination: Now I solve systems of linear equations with elimination |
| 34: Vectors & Linear Equations XIX |
Live #34 |
Youtube #34 |
Problems #34 |
Notes #34 |
The Goals of Elimination: There are exactly four specific goals to Elimination. Now I outline them all. |
| 35: Vectors & Linear Equations XIX |
Live #35 |
Youtube #35 |
Problems #35 |
Notes #35 |
The Goals of Elimination II: Once again, I outline all the goals of Elimination. |
| 36: Vectors & Linear Equations XIX |
Live #36 |
Youtube #36 |
Problems #36 |
Notes #36 |
Switching Rows: Here's something we haven't encountered before: the need to switch rows! |
| 37: Vectors & Linear Equations XIX |
Live #37 |
Youtube #37 |
Problems #37 |
Notes #37 |
Matrix Elimination Notatoin: I finally introduce the actual Elimination Matrices involved in Matrix Elimination. |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
| Linear Algebra |
Youtube |
