122 - מרחב וקטורי: דוגמאות סטנדרטיות

Summary

Join Dr. Aliza Malek as she delves into the world of vector spaces in linear algebra. Learn about the ten critical properties that define vector spaces and explore standard examples using matrices. Discover fascinating parallels between row and column matrices, and how they relate to vector spaces Fn. Examine special cases and combinations of fields, plus mistakes to avoid when mixing them. Whether new to the concepts or looking to deepen your understanding, this engaging lesson sheds light on the intricacies of vector spaces!

Highlights

• Dr. Malek revisits the essentials of vector spaces, laying out the ten must-know properties. 🌟
• Standard examples include matrices and their operations, helping visualize vector spaces in action. 📊
• Special focus on Fn and differences in notation, crucial for understanding linear equations. 🔢
• Insightful discussion on mixing fields and the impact on properties of vector spaces. 🌈
• Fun illustration of how real and complex fields play differently within vector spaces. 🎨

Key Takeaways

• Vector spaces have ten essential properties that make them the foundation of linear algebra. 📚
• Matrix examples provide a practical understanding of vector spaces in various dimensions. 👩‍🏫
• Mixing different fields can result in non-standard vector spaces; always check the field compatibility! 🔄
• R2 and R3 represent Euclidean planes and spaces, familiar to anyone who's done basic geometry. 🦾
• Knowing the distinction between rows and columns is crucial to avoid notational errors in matrix operations. 📐

Overview

Dr. Aliza Malek is back with another riveting session on vector spaces, where she revisits the fundamentals of these mathematical constructs. Vector spaces, defined with their ten crucial properties, present a structure where the rules of linear algebra come to life. It’s a bustling world where abstract ideas turn into calculable elements, perfect for those eager to explore mathematical dimensions!

In this lesson, Dr. Malek walks through standard examples like matrices of various orders, underlining their role as vector spaces over fields. With clarity and precision, she emphasizes the flexibility of these spaces in theoretical applications, coupled with a practical look at distinguishing row and column matrices to avoid common errors.

The session wraps up with an enlightening dive into the fusion of real and complex fields, accompanied by a quirky mascot to reinforce learning. Avoiding pitfalls while mixing fields ensures proper scalar multiplication and the preservation of vector space properties. This overview leaves you eager for the next lesson, promising more fun with standard space groups!

Chapters

• 00:00 - 00:30: Vector Spaces - Standard Examples This chapter, part of a Linear Algebra course taught by Dr. Aliza Malek, focuses on standard examples of vector spaces. It follows a previous lesson that introduced vector spaces over a field F. The chapter begins by reiterating the definition of a vector space, which involves a group V and a field F, setting the stage for exploring more examples in the realm of linear algebra.
• 00:30 - 01:00: Properties of a Vector Space This chapter discusses the properties of a vector space. It explains that in a group V, we can perform addition and provide scalars from field F to multiply terms. V is considered a vector space over field F if certain conditions are met. Specifically, for any three vectors in V and two scalars in F, a series of ten attributes must be satisfied. These conditions define the structure and behavior of vector spaces and their interaction with scalars from their respective fields.
• 01:00 - 01:30: Review of Vector Space Properties The chapter discusses the various properties of vector spaces. It emphasizes that there are ten key properties to consider, which include closure, associativity, commutativity, indifference, and counter terms. The instructor highlights that these attributes are fundamental and logical to expect in the context of vector spaces. There is no mention of any unusual or strange properties; rather, the properties are deemed natural and necessary for the mathematical framework being discussed.
• 01:30 - 02:00: Matrices as Vector Spaces In this chapter titled 'Matrices as Vector Spaces', the concept of matrices as vector spaces is discussed. It starts by considering matrices of order m by n with elements in field F, observing them as a vector space over field F. The chapter emphasizes on usual scalar addition and multiplication in matrices as the actions. It also highlights the importance of reviewing other standard examples which will be integral throughout the course. Furthermore, it describes that whenever a theory is formulated, the discussion will involve a vector space V over the field F.
• 02:00 - 02:30: Special Cases of Matrices This chapter, titled 'Special Cases of Matrices,' introduces the concept of special matrix forms, specifically focusing on row matrices and column matrices. It explains that a row matrix is a matrix with dimensions 1 by n, meaning it has only one row, and a column matrix has dimensions n by 1, meaning it has only one column. The chapter also refers to previous and future lessons for a more comprehensive understanding of matrices.
• 02:30 - 03:00: Representation of Matrices as Fn This chapter discusses the concept of representing matrices in a way that does not distinguish between row and column matrices as long as they have the same number of terms. The focus is on eliminating the difference between 1 by something and something by 1 dimensions by focusing on the number of terms, thereby treating them equivalently.
• 03:00 - 03:30: Maintaining Notation Rules In this chapter titled 'Maintaining Notation Rules,' the focus is on understanding the representation of matrices, specifically the notation used for matrices referred to as 'Fn.' The chapter discusses the flexibility of notation to avoid specifying whether a matrix is 1 by n or n by 1, simply referring to it as 'Fn.' It introduces the concept of matrices consisting of a single column with n terms (a1, a2, ..., an), where each term belongs to a set F. The idea emphasized is that these column matrices maintain the same elements as rows, ensuring consistency in notation and representation across different forms.
• 03:30 - 04:00: Examples of Matrices in Fn The chapter focuses on the concept of matrices in the context of a group referred to as Fn, where rows and columns are treated equivalently. The notation Fn signifies a matrix with equal number of rows and columns. However, when specific differentiation between rows and columns is necessary, the notation n by 1 or 1 by n is used. A significant emphasis is placed on adhering to notation rules to avoid confusion.
• 04:00 - 04:30: Real-World Applications of Matrices This chapter discusses the real-world applications of matrices, specifically in the context of a system of linear equations. It introduces matrix A with order m over n and terms in field F, and explains how it is used to construct equations in the form Ax=b. The focus is on understanding the notation and structure, emphasizing that even though we write Fn and may not distinguish between rows and columns, the rules of notation must still be adhered to.
• 04:30 - 05:00: Euclidean Examples of Vector Spaces The chapter discusses Euclidean examples of vector spaces, focusing on the expression Ax=b. It emphasizes the importance of consistent matrix and vector dimensions for multiplication to be defined, specifying that A is an m by n matrix and x must be a column vector in Fn. It highlights the necessity of writing vectors as columns even though rows and columns are inherently similar.
• 05:00 - 05:30: Vector Space Over Itself This chapter, titled 'Vector Space Over Itself', discusses the result of matrix multiplication. It explains that for matrices of sizes m by n and n by 1, the product matrix will be of size m by 1, implying that the matrix 'b' must be a column vector. The chapter emphasizes the importance of maintaining proper notation rules when dealing with matrices. It also briefly touches on the general solution of a matrix system, stating that the solution can be expressed as a line and is independent of matrix multiplication.
• 05:30 - 06:00: Complex Numbers in Vector Spaces The chapter explores the use of complex numbers within vector spaces, particularly focusing on solving the system of equations represented as Ax=0. By using a 2 by 2 matrix A, the chapter illustrates how to define multiplication in such systems. With the x column being 2 by 1, the resultant vector is also 2 by 1, thereby ensuring the system's outputs are compatible in size and structure. The solution space encompasses vectors in R2 or C2, highlighting the flexibility of using real or complex numbers in the analysis of such mathematical systems. This approach reinforces the understanding that the zero vector, here termed as 0,0, exists within these frameworks, denoted as F2.
• 06:00 - 06:30: Mixing Fields in Vector Spaces The chapter titled 'Mixing Fields in Vector Spaces' discusses the representation of solutions in vector spaces. It highlights that traditionally, solutions are presented as columns to maintain consistency with input representation. However, the chapter explains that for certain mathematical operations and solutions, especially when they are apparent or trivial, expressing the solution as a line is permissible. This flexibility is justified by the clear understanding of the relationships and equations in the system, demonstrated by examples such as x-y=0 and 2x-2y=0, showing how solutions can diverge from the traditional column representation.
• 06:30 - 07:00: Properties of Mixed Fields This chapter covers the properties of mixed fields, particularly in the context of matrix operations in a 2-dimensional space (R2). It emphasizes the importance of differentiating between R2 and a higher dimension like R2 by 1, where the arrangement or operations on rows and columns differ significantly. The chapter underscores the necessity of adhering to established mathematical rules of writing and analyzing matrices. It exemplifies using a matrix A with rows (1, -1) and (2, -2), prompting readers to determine their specific locations or statements within a mathematical framework.
• 07:00 - 07:30: Validity of Matrix Operations This chapter discusses the validity of matrix operations, specifically focusing on square matrices and their properties. It highlights how the rows and columns of a square matrix belong to the same vector space Fn, using R2 as an example. The chapter also cautions against improperly combining matrices by mixing rows and columns in calculations, emphasizing careful adherence to proper matrix operation rules.
• 07:30 - 08:00: Complexity in Scalar Multiplication The chapter discusses the concept of complexity in scalar multiplication of matrices. It clarifies that while rows and columns are essentially equivalent, when adding matrices, they must be of the same order to avoid errors. The explanation uses an example of adding two columns (1,2 and 3,4) to demonstrate that they result in a new column (4,6), emphasizing the importance of maintaining order to prevent spelling errors in matrix operations.
• 08:00 - 08:30: Combinations of Fields and Spaces This chapter focuses on combinations of fields and spaces in relation to matrices and arithmetic operations. It emphasizes the importance of correct calculation and representation of matrices, noting that rows and columns can be perceived similarly when arithmetic is not involved. The example given involves understanding the row (4,6) as separate from matrix arithmetic.
• 08:30 - 09:00: Illustration of Field Combinations The chapter titled 'Illustration of Field Combinations' explores specific and well-known instances of mathematical groups denoted as Fn, particularly when the field F is R (Real numbers). The discussion begins with the simplest case where n=2, leading to the familiar coordinate pair notation (a1, a2) being reimagined as (x, y). The text hints at familiarity with these concepts, suggesting prior knowledge or intuitive understanding of the implications of the notation x, y within this mathematical context.
• 09:00 - 09:30: Conclusion and Next Steps This chapter delves into the Euclidean plane concept, emphasizing the ordered pair (x, y) structure. It highlights how vectors in this plane, characterized by their length and direction, represent a fundamental aspect of Euclidean geometry. The discourse distinguishes between R2, the two-dimensional representation, and extends the discussion to n=3 dimensions, offering a more conventional notation for enhanced comprehension. The chapter serves as a review of vector theory and its applications, providing groundwork for further exploration of vector spaces in three dimensions.

122 - מרחב וקטורי: דוגמאות סטנדרטיות Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Vector Spaces - Standard Examples Linear Algebra Course Dr. Aliza Malek Hello, thanks for coming back. In the previous lesson we learned what a vector space over field F is, A world where linear algebra exists. Today we will continue to look at more standard examples, let's get started. First we recall the definition of a vector space. We have a group V, we have a field F.
• 00:30 - 01:00 In the group V we perform addition, F provides us with scalars for multiplying a term from V with a term from field F, with the scalar. V is called a vector space, v.s, over F field, if for every u,v,w in V, every three vectors in the group, for every alpha,beta, two scalars in field F, without exception, all of the following ten attributes are met.
• 01:00 - 01:30 Now we will just go through all ten properties and... At this point, I have to assume we remember these attributes. Note that there is not a strange attribute here. All the attributes are things that are very natural to ask for: closure, associativity, commutativity, indifference, counter terms. No property here, brackets opening and else, is all very natural to ask for.
• 01:30 - 02:00 Now let's continue to see the examples. We have already seen matrices of order m by n with elements in the field F, let them be a vector space over field F. Which actions do we choose? Usual scalar addition and multiplication in matrices. Now we will continue to review some other standard examples that will accompany us in the course. What does that mean? Whenever we formulate a theory, we will talk about a vector space V over the field F.
• 02:00 - 02:30 Whenever we want to show examples, we will choose one of the standard examples we will get to know in this lesson, the last lesson, and in the next lesson we will see a few more. So first of all, I now want to show you two special cases of the matrices of order m by n. The first case is row matrices, matrices of order 1 by n, one row. We also have the special case of matrices of order n by 1, column matrices.
• 02:30 - 03:00 Now we want not to distinguish between them, So that as far as we are concerned row matrices and column matrices will be exactly the same, as long as there are n terms. So look at what I will do. What differentiates them? If here I have n columns, and here I have n rows, the only thing different is 1 by something, or something by 1. Let's get rid of the 1.
• 03:00 - 03:30 What are we left with? This is Fn, and this is Fn: Now we don't see whether it's 1 by n, or n by 1, it is just Fn. And here is the new group that we get. Fn will be column matrices, of a single column with n terms. a1, a2, up to "an", while a1, a2, up to "an" are in F, But the columns will be the same as the rows.
• 03:30 - 04:00 And so we look at the group as Fn, rows and columns will be the same for us, and we will just call it Fn. When we want to distinguish between a row and a column, we say n by 1, 1 by n. When we say Fn, rows and columns will be the same. But be careful, it is very important to remember, we must maintain the notation rules.
• 04:00 - 04:30 Although we write Fn and do not distinguish between rows and columns, the rules of notation must be maintained. Let's see what it means. Let's look for example at a matrix in a system of linear equations, we have matrix A, an order m over n with terms in the field F, and we write a system of equations Ax=b. We now look at x as a term in Fn,
• 04:30 - 05:00 and b as a term in Fm. But, when we write Ax=b we must be careful, A is an m by n matrix, if I want the multiplication to be defined, we have no choice, x must be a column. Indeed, x is in Fn, but we must write it as a column although a row and a column are the same thing,
• 05:00 - 05:30 we will simply have a typo, and we don't want that. What will be the result? m by n, n by 1, the result is m by 1, so this b must be a column. We will maintain the notation rules. The general solution, when we finish solving this system, we will be able to write the solution as a line, because it is already disconnected from the multiplication of matrices we see here.
• 05:30 - 06:00 Let's see an example. I am looking at the system Ax=0, and we want the general solution. A 2 by 2 matrix, and for the multiplication to be defined, the x column is 2 by 1. The result of the multiplication is 2 by 1, so b is written as 2 by 1. x,y is in R2, or C2, doesn't matter, 0,0 is in F2,
• 06:00 - 06:30 but is written as columns, so that the multiplication will be defined. But, when we solve the system, it is very easy to understand what is happening here, we have x-y=0, 2x-2y=0, the solution is very clear to us already, x,y equals to x,x. We are allowed to write it as lines. Up until today we always made sure to write x,y as a column, why? Because x,y is given as a column. What allows me to write the answer as a line?
• 06:30 - 07:00 The fact that we are in R2, and not in R2 by 1. In R2 the rows or columns are the same. But we must maintain the writing rules. Now we can also look at this matrix A, 1, -1, 2, -2, and say: its rows, 1, -1, and the second row 2, -2, are where exactly?
• 07:00 - 07:30 In R2, and not in R1 by 2. Now we can also say that its columns, 1,2,-1,-2, instead of saying that they are in R2 by 1, we say that they are also in R2. That is, when I have a square matrix, its rows and its columns are in the same world, Fn. Here is another thing to be careful about. We must not write 1,2 as a column, plus 3,4 as a row.
• 07:30 - 08:00 But we said that rows and columns are the same thing? True, but we also said that to add matrices they have to be of the same order, and so we won't write an addition this way. What will we write? We will choose rows, or columns, it doesn't matter which, the main thing is that it is in the same order as we write it, so that we do not have a spelling error. 1,2 as a column, plus 3,4 as a column, gives 4,6 as a column.
• 08:00 - 08:30 Now, I can conclude and say, we got 4,6, and we write it as a row. Because this row, 4,6, is disconnected from the matrix arithmetic we did here. Just make sure to calculate correctly, and write the matrices correctly when doing arithmetic. When you don't do arithmetic, a row and column will be the same.
• 08:30 - 09:00 Let's look at some special and well-known cases of Fn, the new group we got to know, where F is the field R. The first case, n=2, let's see what we get. R2, instead of saying a1,a2, let's say something familiar, x,y. Wait, we know what x,y is,
• 09:00 - 09:30 it is a collection of all pairs of the form x,y, ordered pairs. Why, this is the Euclidean plane, right... And notice that in the Euclidean plane we have the arrows with the length and the direction. This is the special case of arrows with length and direction, it's simply R2. So you probably also know the special case where n=3. Instead of writing a1,a2,a3, let's write it in the more familiar way.
• 09:30 - 10:00 R3 is all terms of the form x,y,z, where x,y,z are real numbers. What is this? Right, this is the Euclidean space. These are the arrows in space that we know, with length and direction. A special case of a vector space. Wait, we can also have n to be 1, then we have R1. R1, but R1 is actually R. R, all the x's, so that x belongs to R, is basically the straight line,
• 10:00 - 10:30 R1 is the straight line. Wait, but R is the field, and therefore, every field is a vector space above itself. When the addition within R will be the normal addition of the field, and multiplying by a scalar, I take a term from R, the field, and multiply by a term from R the group,
• 10:30 - 11:00 and therefore, multiplication by a scalar is in fact the normal multiplication within the group. So every field F is also a vector space over itself, over F. Let's see another special case. This time we will take F to be the complex numbers, and we will take V to be C2, what is C2? Pairs, because it is 2, of numbers within C.
• 11:00 - 11:30 Let's see how this looks. Here is C2, all terms of the form z and w, while z and w are in C, that is how we defined Fn. This time F is simply C. Ordered pairs of complex numbers. But wait, we write complex numbers in a slightly different way, with real numbers. That's right, therefore C2 can also be written in the following way: a+ib, c+id, where a,b,c and d are real numbers.
• 11:30 - 12:00 This is another way of writing C2 above the C field. Our mascot is asking us whether it is the same field all the time? Definitely. In the standard examples we will always stick to the same field. What does the same field mean? Pay attention, When I construct the group V, my terms are in some field C. Here I construct ordered pairs of complex numbers, here I construct ordered triples of real numbers, and so on.
• 12:00 - 12:30 And we also have the field that supplies the scalars. Which field do we choose? The exact same field. In the standard examples it is forbidden to mix between fields. But if you want to mix fields up, you have to be careful. Let's now check what happens when we mix the fields. I like to call it the "combination" example.
• 12:30 - 13:00 Let's do combinations. Why F and F, let's mix fields a bit. So here is the first example. The first mix I want to do will be F=C, V=R2. That is, I construct terms in the group V with real numbers, and when I multiply by a scalar, I will be allowed to multiply them by complex scalars. Because the field F that provides us with scalars, will be a complex field.
• 13:00 - 13:30 Let's see what happens here. We have to check ten properties, remember? Properties one through five, if you recall, only concern addition. The field is not manifested here, and we have already seen that in R2 the properties are fulfilled, this is a special case of the matrices. One through five are fulfilled, it has nothing to do with the field. This is true in R2 over R, it will also be true in R2 over C. Now we get to the properties in which the field is manifested,
• 13:30 - 14:00 properties six trough ten, let's check out six. When I take a real term, ordered pairs of real terms in R2, and I multiply by the scalar C, which not necessarily real, will I stay in the group R2? A good question. I guess you already notice that the answer is no.
• 14:00 - 14:30 Because if I choose v, for example, to be 1,0, and I choose the scalar alpha to be i, I am allowed. Indeed, I'm allowed to choose alpha to be 7, but if I'm in C, alpha is also allowed to be i. What happens now when I do alpha times v, will I stay in R2? So the answer is of course no, alpha v is i times 1,0, when we multiply the same way we multiply in R2, which is like matrix multiplication, i,0.
• 14:30 - 15:00 It's no longer in R2. Property six does not hold. We can stop everything and declare that this is not a vector space. And notice, although it is not a vector space, but properties seven to ten, which are only related to the properties of the scalar multiplication with the matrices, will hold. Only property six does not hold.
• 15:00 - 15:30 Everything holds, seven to ten hold, but outside of R2, so it's not right for us. So this case is not a vector space. Now we will continue with the combinations example, and mix it up a little differently. This time we will take the field to be real, and take the group which consists of complex members. C2, ordered pairs of complex terms,
• 15:30 - 16:00 which we would like to multiply by real scalars only. Let's check out the ten properties. One to five hold, as we mentioned, because it is true in Fn over F for every field F, therefore C2 above C also has no meaning for field F, which in this case is C, or will be R, it doesn't matter. Now we get to properties six to ten where the field is manifested.
• 16:00 - 16:30 Let's start from property six. Now I want to know whether alpha v is within C2, when v consists of complex terms, real or not, it doesn't matter, just complex, and the alpha must be real. What will happen now? A real term times a complex term, what will we get? True, we already know we will get something that is complex, and something that is complex will indeed be found in C2. Therefore, this time the answer is yes.
• 16:30 - 17:00 A complex vector times a real scalar will always remain complex. Property six holds. We already said that properties seven through ten are is not related at all. These are properties of the operations of alpha times v, so they too hold. All ten properties hold, and so pay attention, we got C2 is a vector space over the field R. This combination worked for us.
• 17:00 - 17:30 Wait, but I don't have to take the C2 matrices. I can take matrices in any order I want, For example complex matrices of order 2 by 4, or 7 by 10. No matter which order we take, complex matrices will be a vector space over the field R. Indeed, we can choose a smaller field. If the terms in V are constructed from terms in the field C,
• 17:30 - 18:00 I'm allowed to see it as a vector space over the same field C, that's the standard, or over a smaller field, such as R, or Q. But the field must not be enlarged. Because this one is real, and here the field is complex, and I provide scalars in a larger field, then I get outside of property six. Time to see our illustration, remember?
• 18:00 - 18:30 Here is the field. We will stick with F being a real field. Here are all of our clouds. Let's take a look at the examples we have already seen, and at the new examples we have added this time. So we have R2, real 6 by 6 matrices, real 2 by 3 matrices, R8, which is eight real terms in a row, or in a column, no big deal, it will be the same thing.
• 18:30 - 19:00 What does the field do? Of course, it provides scalars for all. But now we also have C2 as a vector space over R. But remember, we mixed the fields, we made combinations, so it is not a standard vector space, it is a vector space, but not a standard one. If we want to work with it, we will declare that in advance in an orderly manner.
• 19:00 - 19:30 As long as we say C2, it's not that, it's C2 above C. We want C2 over R? Then we must declare in advance, because this is a non-standard example. That's it my friends, now we already have two types of standard examples: Matrices m by n, and Fn. In the next lesson we will get to know another group of standard spaces.
• 19:30 - 20:00 That's it for now, thank you very much.