Lecture1

★ 向量、矩阵及其运算法则是描述、分析、处理线性系统的有力工具——其“有力”具体表现在这种工具的普适性和简便性上。

★ 学习基础知识 → 专业课程中进一步认知 → 科学研究中应用第一讲 线性空间与线性变换一、线性空间的定义及性质 [预备知识]★ 集合：集合是数学中最基本的概念之一。

所谓的集合是指一些事物（或者对象，称为元素）组成的整体。

集合的运算：设S 是任意非空集合，如果对任意,x y S ∈，按某种规则规定了S 中的唯一元素z 与,x y 对应，则称这种规则为集合S 的一个（二元运算）。

关键在于运算的封闭性，即z S ∈。

集合的表示：枚举、表达式，如{1,2,5}A =; {1,2,3,...}A =; 2{0}A x x x =+>元素a 属于集合S ： a S ∈ 元素a 不属于集合S ： a S ∉ 空集： Φ子集：1212()S S a S a S ⊂∈⇒∈真子集：1212,S S S S ⊂≠ 集合相等：1221,S S S S ⊂⊂集合的交集：1212{|}S S x x S x S =∈∈∩且 集合的并集：1212{|}S S x x S x S =∈∈∪或 集合的和集：1212{|}S S x y x S S +=+∈∈,y另外，集合的“和”（+）：并不是严格意义上集合的运算，因为它限定了集合中元素须有可加性。

★ 数域：一种数集，含有非零数，对四则运算封闭（除数不为零）：任意两数的和、差、积、商的结果仍属于该集合。

比如有理数域、实数域（R ）、复数域（C ）、四元数域。

实数域和复数域是工程上较常用的两个数域。

线性空间是线性代数最基本的概念之一，也是学习矩阵理论的重要基础。

线性空间的概念是对各种具体线性系统的一种统一的抽象。

1. 线性空间的定义：设V 是一个非空集合，其元素用,,x y z 等表示；P 是一个数域，其元素用,,k l m 等表示。

如果V 满足（I ）在V 中定义一个“加法”运算，即当,x y V ∈时，有唯一的“和”x y V +∈（封闭性），且加法运算满足下列性质（1）结合律： ()()x y z x y z ++=++； （2）交换律： x y y x +=+；（3）零元律： 存在零元素0，使0x x +=；（4）负元律： 对于任一元素x V ∈，存在一元素y V ∈，使0x y +=，且称y 为x 的负元素，记为()x −。

托福听力tpo63lecture1、2、3原文+题目+答案+译文Lecture1 (1)原文 (1)题目 (3)答案 (5)译文 (5)Lecture2 (7)原文 (7)题目 (9)答案 (11)译文 (11)Lecture3 (13)原文 (13)题目 (15)答案 (17)译文 (17)Lecture1原文Listen to part of a lecture in a geology class.Professor:OK.Before we begin,I wanna remind you that our field trip to Bryce canyon national park is this weekend.Remember the bus leaves early,five am,so don't forget to set your alarm clocks.I think you're all gonna enjoy getting out of the classroom and actually seeing some remarkable geologic phenomena.Now,while we're there,I want you to pay particular attention to two things.One obviously will be the sediment layers making up the rocks,since we've spent so much time onsedimentary rocks.Bryce canyon is a great place to see how millions of years have turned layers and layers of tightly packed sediment,mud particles,sand remains of plants and animals into rock.But you're also gonna see some fascinating rock shapes, formations that are the result of the weathering and the erosion processes that occur at Bryce canyon.There are two main processes that are important.The first one is a weathering process called frost wedge.Frost wedge a process that widens cracks in rocks in the wintertime.It begins with warm air or daytime sun melting the snow.As the snow turns into water,it seeps into the cracks that occur naturally in sedimentary rocks.At night,this water freezes in the cracks,but when water freezes,it expands quite a bit, which means that it prize cracks open,gradually,making them wider and breaking off a little bits in the process.Now,this thought freeze cycle can happen as many as two hundred times in a single year.So that makes it the most important weathering process at Bryce canyon.The other key processes is runoff,which is an erosion process.Runoff takes place in the summer.The parks in the desert said the grounds very dry.When it rains in late summer,the ground is too hard to absorb the water,so it runs off.And as it runs off, it carries away the gravel,the broken bits of rock created by cross wedge in the winter.So runoff is the main erosion process that alters the rock landscape in the park.And because these processes have occurred over thousands of years,some of the results can be pretty dramatic,like the giant corridors are passage ways that have developed within the rocks.These passage ways are known as slot canyons.Here's an example of one,not from the part we're going to.This one is actually in Australia,but the scale is typical.So these huge spaces started out as small cracks throughout the sedimentary rock,then thanks to millions of cycles across wedge and runoff.What used to be one big area of rock is now sort of two smaller areas of brought with the corridor in between.We'll have a chance to walk through some like this.These slot canyons are great places to explore,but let me just say,for any of you who aren't from around here,if you ever go on your own,make sure you check aweather forecast first.A sudden heavy rain can cause a flash flood in a slot canyon. So you want to know when it's safe to explore them.Unfortunately,it'll be dry this weekend.Now,these deep,narrow slots are pretty common.You might even have two of them very close to each other with only a thin wall of rock in between.Of course,frost wedge is still at work,so it starts wearing away at the front of the thin wall until you get a whole I mean a hole all the way through the wall,front to back.And this hole gets bigger and bigger.Once it's at least one meter in diameter,it's called a window.And eventually the weight on top of it is just too much,so the roof caves in and only the sides,sometimes it's just one side is left standing.These sides,which look a lot like collins,now are called Hudos.Here's a photo of something we'll be seeing.One of the things that makes Bryce canyon unique is that it has more Hudos than anywhere else in the world.Yes,Margot?Female student:Why is it so lumpy looking?You'd think it would be smoother.Professor:Well,remember,these are sedimentary rocks,so they have layers.Some layers are mostly limestone,and limestone erodes pretty quickly in the presence of any kind of acid.Now Bryce canyon in a very unpolluted area,but even,there the rain water has a little carbolic acid in it,which causes the limestone to erode.But other layers are made up of different types of sediment,which aren't so vulnerable to acid,so they don't erode as quickly.题目1.What is the lecture mainly about?A.The length of time required to produce sedimentary rocksB.The role of climate conditions in the creation of sedimentary rocksC.Some processes that produced a specific group of rock formationsD.Some unique geologic features found in canyons in the United States2.According to the professor,what is one characteristic that frost wedging and runoff share?A.Neither occurs in a desert.B.Neither is a frequent event.C.Both are weathering processes.D.Both are seasonal phenomena.3.Why does the professor show a picture of a slot canyon?A.To give students a sense of the size of a typical slot canyonB.To show students one of the places they will visit on their field tripC.To illustrate how many sediment layers are visible in a typical slot canyonD.To show how much slot canyons can vary based on local climate conditions4.What is the professor's attitude toward students exploring Bryce Canyon on their own?A.He worries that students may not know to take appropriate precautions if they go by themselves.B.He suspects that many students will not go on their own if such a trip requires them to get up early.C.He hopes that the class field trip will motivate students to visit Bryce Canyon on their own.D.He believes that students learn more from individual exploration than they dofrom being in a group.5.How is a hoodoo formed?A.Runoff produces large gravel deposits.B.Air pollution leads to a buildup of limestone.C.The roof of a rock window collapses.D.A flash flood washes away the base of a rock wall.6.According to the professor,what two factors explain why a hoodoo does not have a smooth shape?[Click on2answers.]A.The presence of acid in rainwaterB.The temperature swings between the summer and the winter seasonsC.The composition of the hoodoo's sedimentary layersD.The location of the cracks created by frost wedging答案C D A A C AC译文听一段地质学的讲座。

托福听力tpo56 lecture1、2、3 原文+题目+答案+译文Lecture1 (1)原文 (1)题目 (3)答案 (5)译文 (5)Lecture2 (7)原文 (7)题目 (9)答案 (11)译文 (12)Lecture3 (14)原文 (14)题目 (16)答案 (18)译文 (18)Lecture1原文NARRATOR: Listen to part of a lecture in an art history class.MALE PROFESSOR: OK, at the end of our last class I started to talk a little bit about a dominant movement in United States painting in the late 1940s and the 1950s. And I said that the artists involved shared a spirit of revolt against tradition and a belief in spontaneous freedom of expression. This significant art movement is known as Abstract Expressionism.Now, Abstract Expressionism is kind of hard to define, but it-it’s basically an attemptby the artist to convey meaning or feeling in an abstract way. So, the artists didn’t worry about whether they were painting familiar subject matter, like the kinds of things you’d see in the world around you. They’d paint...well, abstract things, on, ah, a huge canvas—which itself was a break from traditional technique. And it was common among artists to apply the paint to the canvas very rapidly and with great force. So let’s look at the work of the most famous American Abstract Expressionist, Jackson Pollock.There was nothing in Jackson Pollock’s training as an artist that suggested he would come to be seen as some sort of artistic revolutionary. In the 1930s he studied drawing and painting at the Art Students League, a popular art school in New York City. What he did later—in the 1940s—was a startling innovation. Jackson Pollock used a technique, the so called “pour and drip” technique, for which he is best known. He didn’t use the traditional easel—he laid his wall-size canvas flat on the floor, so he could move around it and work it from all sides. Then he poured and dripped his paint onto the canvas without touching it with a brush—just poured and dripped.Now, the physical movements involved in Pollock’s painting technique have led people to call it “action painting,” which almost suggests that the process of creating the painting, physically, was at least as important as the end product itself. In fact, people used to watch him work in his studio, dripping and pouring paint and other materials onto his canvases. This could make you think of Pollock’s work as being kind of like, wild or chaotic, or random. But the truth is that Pollock was in complete control of his materials and his paintings.Pollock’s pour and drip works were quite revolutionary, and at first they shocked the art world. Pollock used massive canvases. They seem more like portable murals than anything else. A good example of his technique is the painting “Autumn Rhythm,” which Pollock painted in 1950.“Autumn Rhythm,” at first glance looks like basically, just a whole lot of squiggly lines;rather bizarre, just like a bunch of pointless drips and swirls. But if you look closely, you see why it’s so admired.Beneath all the apparent chaos there’s really a very definite structure of lines, rhythms, and sensations that makes the whole piece work. Sheer randomness would not be nearly as visually appealing as this painting is. You need some structure, even if it’s not readily apparent.I’ve read some articles by other scholars who’ve, in their discussion of Pollock, um, some of them like to point out that he painted his canvases while looking down at them, since they were on the ground, as I said, but when we go to a museum, they’re up on a wall. They think this is significant because it makes our perspective different. But I mean...well, think of photography. We’ve all seen photos of the sky, the ground...meaning that the photographer was shooting from different angles. Does that mean that we should put a photo of the sky, on the ceiling? Of course not. It wouldn’t matter if you’re looking at it on a wall or in a photo album on your lap. And I think it’s the same with Pollock. It doesn’t matter from which angle we view his paintings. It’s OK that he painted on the floor and we look at it on the wall.But in spite of his work being shocking and even misunderstood at first, Pollock’s work became so influential in the development of Abstract Expressionism, that the artistic community started to shift its attention from Paris, which had been the center of the art world, to New York, where Pollock lived and worked. So Pollock’s breakthrough work helped move the focus of contemporary art, and that’s one of the measures of his greatness, really.题目1.Why does the professor discuss Jackson Pollock?A. To point out a common misconception about Abstract ExpressionismB. To help students understand the nature of Abstract ExpressionismC. To compare Pollock’s technique to that of other Abstract Expressionist paintersD. To defend Pollock and the Abstract Expressionists from criticism2.What point does the professor make about Jackson Pollock’s training as an artist?A. It motivated him to rebel against art he claimed was boring.B. It contrasted with the type of art he later created.C. It taught him how to paint using unconventional methods.D. It was very different from the type of training most artists receive.3.What were two features of Jackson Pollock’s painting technique?[Click on 2 answers.]A. He used walls as a painting surface.B. He painted the canvas while it was on the floor.C. He applied paint by pouring or dripping it.D. He allowed visitors at his studio to help with the painting.4.What is the professor’s attitude toward the term “action painting”?A. He thinks it correctly describes Pollack’s painting technique.B. He considers it less appropriate for Pollock than for other Abstract Expressionists.C. He believes that it represents the sense of movement displayed in Pollock’s paintings.D. He is pleased that contemporary critics rarely use the term.5.What feature of Autumn Rhythm does the professor imply is representative of Pollock’s works?A. It symbolizes the passage of time.B. It reveals a lack of control over emotions.C. It combines structure and the appearance of chaos.D. It combines tradition and innovation.6.Why does the professor discuss photography?A. To emphasize how different it is from paintingB. To make a point about its increasing popularity in New York’s art worldC. To show the extent of Pollock’s influenceD. To support his argument about the way people look at Pollock’s paintings答案B B BC A C D译文旁白：在艺术史课上听一部分讲座。

托福听力tpo45 lecture1、2、3、4 原文+题目+答案+译文Lecture1 (2)原文 (2)题目 (4)答案 (6)译文 (6)Lecture2 (7)原文 (7)题目 (10)答案 (12)译文 (12)Lecture3 (15)原文 (15)题目 (17)答案 (19)译文 (19)Lecture4 (21)原文 (21)题目 (24)答案 (26)译文 (26)托福听力tpo45 lecture1、2、3、4 原文+题目+答案+译文Lecture1原文NARRATOR: Listen to part of a lecture in an art history class.MALE PROFESSOR: As I was saying, the Renaissance period—which started in the fourteen hundreds in Europe—the Renaissance was still a pretty religious period. And that's reflected in the artwork of that time.But artists were starting to experiment with a more secular point of view as well—a tendency to also use the natural world as the subject matter for their art.And there were different ways that these natural themes were explored. For instance, many artists would paint portraits, while others—although this was more common in northern Europe—would make landscapes the subject of their works.But today I'd like to consider an influential Italian Renaissance artist, Leon Battista Alberti, who took a slightly different approach.Leon Battista Alberti was a painter, sculptor, architect, musician, poet—very wide-ranging interests—like daVinci or Michelangelo, the sort of guy for whom the term “Renaissance man” was in fact created.Alberti believed that the most important approach for a painter was to capture a story or narrative. Now, as I've indicated, this narrative could be either religious or secular, depending on what the work of art was for. If the work was to be placed in a church then obviously it'd have a religious theme, whereas if it was for someone's home then it could deal with a different subject matter. The exact narrative didn't really matter, so long as it was one that captivated the audience—that held the viewer's attention.So what is actually needed to tell a story?Well, Alberti needed characters, right? Human figures.And he wanted to represent them as realistically as possible to capture the viewer's attention. One way he achieved this was to make use of what's known as the contrapposto pose.A contrapposto pose basically entails showing a slight twist in the body. The shoulders and hips are usually bent in different directions. In other words, if the left shoulder is bent so that it's slightly higher than the right shoulder, then the hips will be bent so that the left side will be slightly lower than the right side. Similarly, in sculptures, most of the weight seems to be on one foot, which also results in this slanted position—making it seem like the figure is about to walk or move. This adds to the realistic aspect of the figure.But there are actually a lot of things that could go wrong in the attempt to create such a pose. You could make a figure’s arms bigger than its legs, or the head too small for the body. Messing up the proportions can leave a figure looking cartoon-like and unnatural. But Alberti had a solution: He encouraged artists to visualize a figure's bones and structure. This would give the artist an idea of the proportions of the figure. From there, Alberti suggested the artist imagine attaching the tendons and muscles, then covering those with flesh and skin.Now, although this method may seem complicated, artists since antiquity have used anatomical observations to try to get the proportions of the human figure as accurate as possible—though obviously not to the degree that Alberti was recommending.Now, in addition to characters, the setting is extremely important, especially when attempting to tell a story realistically. Renaissance artists essentially needed to create a three-dimensional scene on a two-dimensional surface. They accomplished this by the use of perspective—a relatively new idea for artists at the time. In particular, the type of perspective that Alberti advocated was called linear one-point perspective. In fact, Alberti was one of the artists who developed the geometry behind linear one-point perspective.Linear perspective basically consists of drawing straight lines that extend from the forefront of the painting into the background—lines that seem to be parallel to each other, but which actually converge on a single point in the horizon, called the vanishing point. By drawing figures and objects smaller and smaller as the lines get closer together, the artist is able to create depth in a painting. This gives the illusion of a third dimension and makes the work of art more realistic.题目1.What is the lecture mainly about?A. Reasons for the transition from religious to secular themes in Renaissance artB. The disproportionate influence of Italian artists during the Renaissance periodC. Techniques used during the Renaissance to produce realistic works of artD. A comparison of themes in paintings and sculptures during the Renaissance2.What is the professor's opinion of Leon Battista Alberti as an artist?A. Alberti's interests were too diverse for him to succeed in any one field.B. Alberti was ineffective in imposing his own theories on other artists.C. Alberti was a much more skilled artist than da Vinci or Michelangelo.D. Alberti represents the Renaissance ideal of wide-ranging achievement.3.According to the professor, what did Alberti consider to be the most important aspect of a Renaissance painting?A. That it convey an appealing narrativeB. That its figures be posed symmetricallyC. That its theme not be religiousD. That its characters be positioned within a landscape4.Why did some artists begin to use the contrapposto pose?A. To create a cartoon-like effectB. To help viewers identify the main figure in a work of artC. To show the relative sizes of human figuresD. To make human figures appear more natural5.Why does the professor discuss tendons and muscles?A. To emphasize that Alberti's study of anatomy led to his interest in artB. To show the emphasis Alberti placed on using physically fit modelsC. To illustrate the difficulty of maintaining a contrapposto pose in real lifeD. To explain one of Alberti's methods for creating accurate proportions6.Why was the development of linear one-point perspective important to Renaissance artists?A. It helped painters to place figures more symmetrically within their paintings.B. It allowed painters to create an illusion of three dimensions.C. It enabled artists to paint large landscapes for the first time.D. It encouraged artists to take an interest in geometry.答案C D A D D B译文旁白：听一篇艺术史学科讲座。