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Emotions Powerpoint

Course: PSYCHOLOGY 220, Fall 2011
School: University of Toronto
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IS EMOTIONS WHAT AN EMOTION? William James (1884) UNCLEAR! Many answersthough no real consensus and numerous issues with the answers BASIC PROPERTIES OF EMOTIONS Brief Signaled through face, touch, voice Specific Cause Helps achieve goals TYPES OF EMOTIONS Primary appraisal: first basic judgment of whether an event is good or bad for the person If good, approach (the positive emotions) If bad, avoid...

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IS EMOTIONS WHAT AN EMOTION? William James (1884) UNCLEAR! Many answersthough no real consensus and numerous issues with the answers BASIC PROPERTIES OF EMOTIONS Brief Signaled through face, touch, voice Specific Cause Helps achieve goals TYPES OF EMOTIONS Primary appraisal: first basic judgment of whether an event is good or bad for the person If good, approach (the positive emotions) If bad, avoid (the negative emotions) WHAT ABOUT SPECIFIC EMOTIONS? Paul Ekman Founding figure in the study of emotions Brought an evolutionary perspective to emotions (heavily inspired by Darwin) Studied universality of basic emotion expression EKMANS BASIC EMOTIONS DECODING BASIC EMOTIONS Ekman traveled to Papua New Guinea Showed the Fore (tribe from highlands) six basic faces High rates of identifying corresponding emotion ISSUES? Book covers criticisms (and responses) of study design What about the specific emotions? What are the issues there? ISSUES WITH EKMAN BASIC EMOTIONS Only 1 positive emotion? Are these really the emotions people see as basic? What are the properties of these emotions? ANOTHER APPROACH TO THE QUESTION OF BASIC EMOTIONS Prototype approach to emotions Prototypes and Categorization Categories that dont have clear boundaries; rather, examples are either better or worse representations of the prototype of the overall category PROTOTYPES OF EMOTIONS Shaver and colleagues had people rate over 200 emotion words 1 = I would never call this an emotion 4 = I would definitely call this an emotion Examples: anger, hate, fear, love, rage, joy, gladness, regret, pride, ferocity, alarm, eagerness, power, aversion, forgiveness, discomfort, rejection, rapture, isolation, uneasiness Through this approach, aimed to evaluate which emotion terms are most prototypically emotions to laypeople PREDICTIONS ABOUT MOST PROTOTYPICAL EMOTIONS? MOST PROTOTYPICAL EMOTIONS Emotion Term Mean Rating Love 3.94 Anger 3.90 Hate 3.84 Depression 3.83 Fear 3.83 Jealousy 3.81 Happiness 3.77 Passion 3.75 Affection 3.72 Sadness 3.68 Grief 3.65 Rage 3.64 WHAT ABOUT BASIC EMOTIONS? Shaver and colleagues then selected the 135 most prototypic emotion terms Put each term on a separate index card Gave all 135 cards to new participants Asked them to put emotion terms that go togethermake as few or as many piles they like Ekmans Emotions Prototype Emotions Joy Joy Sadness Sadness Anger Anger Fear Fear Surprise Surprise Disgust Love APPROACHES TO EMOTIONS Ekman and Prototype approaches are discrete approaches to emotions Love Fear Joy There are also dimensional approaches to emotions Anger DISCRETE APPROACHES Provides insights into how specific emotions differ Specific emotions as a response to specific situations Problem with discrete approaches? Little to say about similarities across emotions DIMENSIONAL APPROACHES Core dimensions of appraisal, when combined, give rise to specific emotions SMITH AND ELLSWORTH (1985) MODEL Dimension Description Attention degree to which you focus on and think about the stimulus Certainty degree to which you are certain about what is going to happen Control coping extent to which you have control over outcomes in the environment Pleasantness degree that the event is positive or negative Perceived obstacles extent to which the pursuit of your goal is blocked Responsibility extent to which other people, you, and situational factors are responsible for event Legitimacy extent to which the event is fair and deserved or unfair and undeserved Anticipated effort extent to which you must expend energy to respond to the events USING DIMENSIONAL APPROACH Interest increased pleasantness the desire to attend the sense that situational factors are producing events a perceived need to expend effort moderate certainty about future outcomes little sense of obstacles or illegitimacy of events DIFFERENTIATING BETWEEN EMOTIONS USING DIMENSIONAL APPROACH Anger Sadness Guilt Agency (combination of control and responsibility) Blaming others produces anger Believing that the situation is responsible produces sadness Self-blame leads to guilt SUMMARY Emotions tend be (a) brief, (b) caused by specific events, (c) have non-verbal signature, (d) goal-directed Primary appraisal: is the event good (positive emotions) or bad (negative emotions)? Basic emotions? Two different answers (though there are others) Ekman: Joy, Sadness, Fear, Surprise, Anger, and Disgust Prototype: Joy, Sadness, Fear, Surprise, Anger, and Love Approaches to Emotions Discrete approaches: view emotions as separate, distinct entities Dimensional approaches: different emotions are a result of different combinations of values on appraisal dimensions UNIVERSALITY VERSUS CULTURAL VARIABILITY Really evolutionary views versus social-constructivist views of emotions AREA OF AGREEMENT Emotions are solutions to basic problems of social living AREA OF DISAGREEMENT Evolutionary framework: emotions are genetically encoded biological processes that emerged in evolution as adaptations to problems or opportunities specific to the environment of evolutionary adaptedness. Emotions are species-characteristic patterns of action and therefore universal Constructionist framework: emotions are words, concepts, representations, and metaphors. Emotions are forms of discourse that emerge within culturally specific institutions Emotions reflect culturally specific concerns about identity, morality, and social structure PUT ANOTHER WAY Emotions in DNA? Emotions a result of culture? AS WITH VIRTUALLY ALL SUCH QUESTIONS IN PSYCHOLOGY Evidence supports both views FOUR WAYS TO VIEW THE DEBATE Potential Versus Practice Prototypes Focal and Variations Emotions within Cultures Idealization POTENTIAL VERSUS PRACTICE Evolutionists focus on potential of emotions Questions are about controlled, in the lab ability Given a highly controlled stimulus, do people of different cultures show similar experiences, expressions, and physiological responses? POTENTIAL VERSUS PRACTICE Constructivists focus on practice How do emotions arise, and how are they experienced and expressed in daily living? POTENTIAL VERSUS PRACTICE At the most abstract level, the appraised antecedents of emotion are similar across cultures Example: Ekman studies of emotion displays Cultural variations in occurs with specific events that trigger emotions that vary by culture Example: In East Asian societies, anger is thought to be highly muted. In the Ifaluk, anger is a highly public, dramatized display PROTOTYPES AND VARIATIONS Emotion have both central and peripheral features Central features: reliably occur with an experience of a specific emotion Peripheral features: less systematically occur with the emotion PROTOTYPES AND VARIATIONS Display of anger Central: furrowed brow, glare, and tightened and pressed lips Peripheral: tightened lower eyelid PROTOTYPES AND VARIATIONS Display of embarrassment Central: gaze down, head turns and movements down, and controlled smile Peripheral: face touches, head shakes, and shoulder shrugs PROTOTYPES AND VARIATIONS Evidence suggests that central traits of emotions are universal Peripheral traits are more culturally variable FOCAL EMOTIONS WITHIN CULTURES Cultures vary in which emotions are focal (emotions that are often experienced) Example: Cultures that emphasize honor (such as Spain), emotions associated with honor (shame, anger) are more focal IDEALIZATION Cultures idealize different emotions IDEALIZATION In the West, emotions such as excitement and enthusiasm are highly valued In the East, emotions such as contentedness and modesty are more valued SUMMARY Emotions can be seen as both universal and highly cultural variable depending on how you look at it People seem to share the potential to experience the same emotions, but in practice they do not Central features of emotions are more universal, while peripheral features are more variable Cultures vary in the emotions they focus on and the emotions the idealize LOVE: A CASE STUDY OF AN EMOTION A FEW INITIAL POINTS Love the most prototypical emotion? Shaver et al 1987 suggests yes! Received little focus until the 1970s Leave it to the poets and novelists WHAT IS LOVE? ANSWERS FROM PSYCHOLOGISTS Little agreement! Emotion? Motivation? Action tendency? Attitude? WHAT WEVE LEARNED ABOUT LOVE Romantic Love is a world-wide phenomenon WHAT WEVE LEARNED ABOUT LOVE Highly variable across people and cultures TYPES OF ROMANTIC LOVE Many different typologies developed over the years Passionate versus companionate Triangular Theory of Love Love Styles PASSIONATE AND COMPANIONATE LOVE Passionate: intense, emotion-filled state of desire for another Companionate: deep affection for another person with whom ones life is interconnected TRIANGULAR THEORY OF LOVE Infatuation Intimacy Companionate Romantic Consummate Passion Infatuation Commitment Fatuous Empty LOVE STYLES Eros: passionate/intense love Storge: friendship-based love Ludus: love as a game Agape: selfless style Pragma: pragmatic love Mania: love alternating between ecstasy and agony LESSONS FROM DIFFERENT TYPOLOGIES OF LOVE? Many different ways to categorize romantic love Love can have high levels of passion, but it need not be Little agreement on the different flavors of romantic love UNIVERSALITY VERSUS CULTURAL VARIABILITY IN LOVE EVOLUTIONARY TAKE ON LOVE Love as solutions to problems Like other emotions, love is functional EVOLUTIONARY TAKE ON LOVE Passionate love solves the attraction problem by motivating individuals to enter monogamous relationships EVOLUTIONARY TAKE ON LOVE Companionate love can be seen as solving the commitment problem in intimate relationships: Maintaining the relationship in the face of other attractive alternatives CONSTRUCTIVISTS TAKE ON LOVE Love varies culturally in many ways CONSTRUCTIVISTS TAKE ON LOVE Individuals with an East Asian background see love as more friendship-oriented than do their Western counterparts CONSTRUCTIVISTS TAKE ON LOVE 50% of Indian participants were willing to marry without love Only 2% of the Japanese participants and 4% of the North American participants were willing to do the same CONSTRUCTIVISTS TAKE ON LOVE Highly individualistic people tend to report lower rates of falling in love than do more collectivistic people Western countries = More individualistic Eastern countries = More collectivistic CONSTRUCTIVISTS TAKE ON LOVE What matters when falling in love? North Americans Appearance Familiarity Readiness Similarity Chinese Reciprocal liking Personality Filling needs Social Influence ANSWERING DIFFERENCE QUESTIONS Evolutionary approaches try to answer why love exists in general Constructivist approaches try to tackle how love varies across cultures WHAT ARE THE LIMITATIONS OF RESEARCH ON LOVE? ONE MAJOR ISSUE Virtually all research is on romantic love But is there more? PROTOTYPE STUDIES OF LOVE First studies (1988): Focused on features of love Second studies (1991): Focused on different types of love FEATURES OF LOVEFREE LISTED Feature % of Subjects Free Listed Caring 44% Happiness 29% Want to be with other 28% Friendship 23% Feel free to talk about anything 20% Warm other feelings 17% Accept the way s/he is 16% Trust 15% Commitment 14% Sharing 14% Think about the other all the time 14% FEATURES OF LOVEPROTOTYPE RATINGS Feature Mean Ratings Trust 7.5 Caring 7.3 Honesty 7.2 Friendship 7.1 Respect 7.0 Concern for the others well-being 7.0 Loyalty 7.0 Commitment 7.0 Accept other the way s/he is 6.8 Supportiveness 6.8 Want to be with the other 6.8 FEATURES OF LOVE When asked to list features of love, concepts such as caring, happiness, wanting to be with other, and friendship arose most frequently More passion related concepts were listed far less often TYPES OF LOVEFREE LISTING Feature # of Subjects Free Listed Feature # of Subjects Free Listed Friendship 51 Lust 14 Sexual 25 Infatuation 13 Parental 23 Of pets 10 Brotherly 22 Physical 10 Sibling 22 True 10 Maternal 20 Emotional 9 Passionate 19 Platonic 9 Romantic 19 Respectful 9 Familial 18 Sisterly 9 Puppy 18 Everlasting 8 Paternal 17 Of life 8 TYPES OF LOVEPROTOTYPE RATINGS Feature Mean Ratings Feature Mean Ratings Maternal 5.39 Spiritual love 4.27 Parental 5.22 Passionate love 4.00 Friendship 4.96 Platonic love 3.98 Sisterly 4.84 Self-love 3.79 Romantic 4.76 Sexual love 3.76 Brotherly 4.74 Patriotic love 3.21 Familial 4.74 Love of work 3.14 Sibling 4.73 Puppy love 2.98 Affection 4.60 Infatuation 2.42 Committed 4.47 Love for humanity 4.42 TYPES OF LOVE Many types of love! Types of love dealing with people (parents, siblings, friends, romantic partners) were most listed and rated as most prototypical Many other types of love (of pets, self love, of humanity) were also listed BASIC PROPERTIES OF EMOTIONS Brief Signaled through face, touch, voice Specific Cause Helps achieve goals DOES LOVE FIT OUR DEFINITION OF AN EMOTION? Is love brief? Hopefully not! Specific causes? Yes Signaled through face, touch, voice? Maybebut certainly different types of love signaled differently Helps achieve goals Likely, though much more research is necessary to figure out those goals TAKE HOME LESSONS No great answer to the starting question: What is an emotion? TAKE HOME LESSONS Most prototypical emotion (love) fails the requirement of what is an emotion!?!? TAKE HOME LESSONS Great complexity within specific emotions that still needs to be investigated TAKE HOME LESSONS Emerging literature on positive emotions 20 years ago, it was all just happiness! MORALITY TWO VIEWS Emmanuel Kant Morality is driven by reason; emotions are secondary David Hume Morality is driven by emotions; reason is secondary WHICH VIEW MAKES MOST SENSE? KANTS VIEW: LAWRENCE KOHLBERG AND STAGES OF MORAL DEVELOPMENT Kants view held court until 1990s Lawrence Kohlberg studied the development of moral reasoning with children Two Central Tenets Reasons guides morality Morals are universal KOHLBERGS MORAL DILEMMAS In Europe, a woman was near death from a special kind of cancer. There was one drug that the doctors thought might save her. It was a form of radium that a druggist in the same town had recently discovered. The drug was expensive to make, but the druggist was charging ten times what the drug cost him to make. He paid $200 for the radium and charged $2,000 for a small dose of the drug. The sick woman's husband, Heinz, went to everyone he knew to borrow the money, but he could only get together about $ 1,000 which is half of what it cost. He told the druggist that his wife was dying and asked him to sell it cheaper or let him pay later. But the druggist said: "No, I discovered the drug and I'm going to make money from it." So Heinz got desperate and broke into the man's store to steal the drug-for his wife. Should the husband have done that? KOHLBERGS MORAL DILEMMAS Asked these dilemmas to children of various ages Not important whether the child say yes or no What was important was the reasoning behind it MORAL STAGES Level 1: Preconventional Morality Stage 1. Obedience and Punishment Orientation. Stage 2. Individualism and Exchange. Level II: Conventional Morality Stage 3. Good Interpersonal Relationships. Stage 4. Maintaining the Social Order. Level III: Postconventional Morality Stage 5. Social Contract and Individual Rights. Stage 6. Universal Principles. Stage 1: Obedience and Punishment Orientation Children obey rules of authorities (usually parents) unquestioningly For Heinz: Typical responses are "It's against the law," or "It's bad to steal, making no other consideration Stage 2: Individualism and Exchange Children recognize there is not just one right or wrong view Everything is relativeso each person is free to pursue their interest But fair exchange acts as an overarching guide. For Heinz: Children stated, Heinz might think its right to take the drug, the druggist would not. Stage 3: Good Interpersonal Relationships Children believe that individuals should act in good ways, live up to the expectations of the community Good behavior is conceptualized as prosociality: motivated to care for others and feel associated emotions (love, empathy, trust) Stage 3: Good Interpersonal Relationships For Heinz: Children suggest that Heinz was in the right to steal the medicine since "He was a good man for wanting to save her," and "His intentions were good, that of saving the life of someone he loves." Stage 4: Maintaining the Social Order. Stage 3 works best with two-person situations In stage 4, being a good person is generalized to society as a whole Stage 4: Maintaining the Social Order. Done through obeying laws, respecting authority, and acting in accordance with promoting social order For Heinz: People understand that Heinz motives were good, but he still violated the law. If everyone broke the law every time there was a good reason, there would be chaos. Stage 5: Social Contract and Individual Rights The question of what makes a good society comes into focus People step away from their own societies to consider what rights and values should societies uphold Stage 5: Social Contract and Individual Rights Believe a good society is best conceived as a social contract in which people freely enter to work toward the benefit of all members of society Believe all rational people should agree on two points: All people want protection of basic rightsliberty and life All people want democratic procedures for changing unfair laws Stage 5: Social Contract and Individual Rights For Heinz: Respondents indicate they dont normally support breaking laws, but that the wifes right to live is a moral right that must be protected STAGE 6: UNIVERSAL PRINCIPLES At this stage, people emphasize the universality of principles such as justice Require principles to be applied equally Actions should be taken with justice for all considered STAGE 6: UNIVERSAL PRINCIPLES For Heinz: All parties druggist, Heinz, and his wifetake the perspective of the other to consider what is most fair. If the druggist did this, the reasoning goes, he would see that life must take priority over property and thus agree that the wifes life must be saved ISSUES WITH KOHLBERGS MODEL? THREE BIG ISSUES No role for emotions? Only morality about harm and fairnesswhat about other questions of morality? Morality really universal (i.e. the same) across societies? EMOTIONS ENTER MORALITY KANTS TIME: JONATHAN HAIDT AND SOCIAL INTUITIONIST THEORY Emotions research and evolutionary theory increasingly prominent players in social psychology Haidts (2001) models central claim: Emotions guide moral judgment Reason justifies the decision after it has been made SOCIAL INTUITIONIST THEORY Emotions are the judge: they make the decisions Reason is the lawyer: it builds the case for why a decision has been made EVIDENCE FOR EMOTIONAL PRIMACY IN MORAL JUDGMENT PEOPLE MAKE QUICK JUDGMENTS OF OTHERS Diverse areas of study have shown that people make quick, intuitive judgments about social objects Such judgments are often driven by emotions MORAL JUDGMENTS AND THE BRAIN Greene and colleagues (2001) showed people different moral dilemmas in an fMRI scanner Example: A train is speeding toward 5 people trapped on a footbridge. You can save them by pushing a fat man in the path of the train, thus using his life to save 5 others. Would you do this? MORAL JUDGMENTS AND THE BRAIN Peoples emotional centers of the brain (ventromedial prefrontal cortex and amygdala) are highly active when reasoning about such dilemmas PSYCHOPATHS HAVE EMOTIONAL DEFICITS Less than 1% of the population are psychopaths, yet they commit more than 50% of the most serious crimes Psychopathy is seen as a moral emotions deficit PSYCHOPATHS HAVE EMOTIONAL DEFICITS Reduced VMPFC, amygdala, insula activity (decreased emotionality) Have emotions but difficulty caring about the fates of other people THE UNEXPLAINABLE MORAL JUDGMENT A man purchases a dead chicken at the market. Once he returns home, he has sex with it. Afterwards, he cleans it, cooks it, and enjoys it for dinner. THE UNEXPLAINABLE MORAL JUDGMENT Sally and Mark are brother and sister. They are traveling through Europe one summer after college. One night on the trip, they both agree it would be interesting and fun to try making love. Sally is already using birth control, but to be extra safe, Mark wears a condom. They both enjoy the experience, but decide to not do it againto keep the night a special memory between the two of them. THE UNEXPLAINABLE MORAL JUDGMENT Many people respond that such acts are immoral When pressed for why, many fail to provide good rationale for their judgment MORALITY ACROSS CULTURES MORE THAN JUST FAIRNESS AND JUSTICE? Five Moral Foundations Harm/care Fairness/reciprocity Ingroup/loyalty Authority/respect Purity/sanctity HARM/CARE Focuses on kindness, gentleness, and nurturance Promotes care and prevents harm Fairness/Reciprocity Focuses on justice, rights, and autonomy Ingroup/Loyalty Focuses on patriotism and sacrifice for the group Authority/Respect Focuses on respecting elders, authority figures, and traditions Purity/Sanctity Views the body as sacred and an entity that should be respected and not defiled. CULTURAL DIFFERENCES IN FOUNDATIONS Cultures can emphasize each moral foundation to varying degrees (if at all!) Example: Domain of authority/respect very important in East Asian societies, but less emphasized in Western Societies ISSUES WITH EMOTIONS FOCUSED, CULTURALLY-SENSITIVE APPROACH TO MORALITY? CURRENT ISSUES AND FUTURE DIRECTIONS Models exist that recognize value of both reason and emotion Example: Joshua Greenes dual systems models Much of the theory behind the Social Intuitionist Theory and Moral Foundations Theory is predicated on evolutionary psychology Evolutionary psychology remains highly controversial within the field Cross-cultural work on morality within psychology is still lacking RIDDLE! You have 100 coins 10 are TAILS up 90 are HEADS up Without looking, how can you make two piles of coins that have equal number of coins facing TAILS up. Piles dont need to be equal You cannot feel the coin faces You cannot look at the coins until you are done No probabilitymust be 100% sure the two piles have equal number of coins TAILS up
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PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
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PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn Linear Algebra or needing a refresher. These notes
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University.Despite the fact that these are my class notes, they should be accessible to anyone wanting tolearn Linear Algebra or needing a refresher.These notes do
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn Linear Algebra or needing a refresher. These notes
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn Linear Algebra or needing a refresher. These notes
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn Linear Algebra or needing a refresher. These notes
Lamar - MATH - 2318
PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn Linear Algebra or needing a refresher. These notes