Linear Algebra 1: Seeking Advice for Midterm Recovery

[Physics-UG]

Hey guys, I'm hoping to get some advice on how to approach this course I'm currently enrolled in.

I'm currently enrolled in linear algebra 1, while I think they're pretty generic for outlines, this is what my school says:

Systems of linear equations. Vectors in n-space, vector equations of lines and planes. Matrix algebra, inverses and invertibility. Introduction to linear transformations. Subspaces of n-space. Determinants. Introduction to eigenvalues and eigenvectors. The dot product and orthogonality. Applications in a variety of fields, numerical methods

Before the midterm I was doing the assignments and, for the most part, they made sense. I didn't really do textbook based problems because it felt like there was a huge leap between the 2 and the textbook was subpar when it came to it's usefulness. I could look at certain things, say orthogonality, and could without a doubt explain it and show it, but with more complex terms/concepts, even though I felt the same way about what it meant, I would have trouble showing it. We just got our midterm back and I didn't do so hot. I didn't fail, but definitely did not do well.

I was hoping you guys could chime in on what you recommend I do to try and bring my grade up a little bit? Could more practice problems help, or is it possible that I'm just not actually understanding the material and should seek a tutor?

Thanks in advance!

 

[micromass]

I didn't really do textbook based problems

I see an obvious room for improvement there! OK, sure, the textbook is subpar. But there are other textbooks out there that would have more decent problems. For example, the following one is free and definitely not subpar: http://www.math.brown.edu/~treil/papers/LADW/LADW.html

We just got our midterm back and I didn't do so hot.

Can you explain what went wrong?

 

[Physics-UG]

I see an obvious room for improvement there! OK, sure, the textbook is subpar. But there are other textbooks out there that would have more decent problems. For example, the following one is free and definitely not subpar: http://www.math.brown.edu/~treil/papers/LADW/LADW.html

Can you explain what went wrong?

I'll definitely be looking at that textbook.

I think it's a lack of real understanding for the foundation of what we've learned. I understand the generic "properties" of such things, say linear independence. But when it's posed in a question I seem to stumble over how to go about the problems. Wording seems to be tripping me up.

 

[micromass]

I think it's a lack of real understanding for the foundation of what we've learned. I understand the generic "properties" of such things, say linear independence. But when it's posed in a question I seem to stumble over how to go about the problems. Wording seems to be tripping me up.

This is not uncommon for linear algebra. Have you tried forming a study group with your fellow students? Trying to figure out things together is definitely beneficial.

 

[QuantumCurt]

To reiterate the above...

I didn't really do textbook based problems because it felt like there was a huge leap between the 2 and the textbook was subpar when it came to it's usefulness.

This is a problem. The best way to learn math is by doing math. People sometimes seem to have an impression that concepts and methods/applications of concepts are learned differently, or are somehow distinct from one another. Sometimes when I'm struggling with qualitatively understanding a concept, I'll simply take for granted that it is true. Then while applying this concept to solve a problem, I'll often have to think about the problem in such a way that makes the concept itself more clear. I try to work as many exercises as I can. I never work enough of them though.

A subpar textbook can certainly be a hindrance. However, one can always get an additional textbook or find supplemental material online.

 

[Physics-UG]

This is not uncommon for linear algebra. Have you tried forming a study group with your fellow students? Trying to figure out things together is definitely beneficial.

I have one friend who I talk with about it, and that's the problem, when we talk I'm like "oh, this makes sense, no worries" and, evidently, that wasn't the case. I think I'll just try and go more into depth with the concepts.

To reiterate the above...
This is a problem. The best way to learn math is by doing math. People sometimes seem to have an impression that concepts and methods/applications of concepts are learned differently, or are somehow distinct from one another. Sometimes when I'm struggling with qualitatively understanding a concept, I'll simply take for granted that it is true. Then while applying this concept to solve a problem, I'll often have to think about the problem in such a way that makes the concept itself more clear. I try to work as many exercises as I can. I never work enough of them though.

A subpar textbook can certainly be a hindrance. However, one can always get an additional textbook or find supplemental material online.

I'll definitely need to do more exercises, the textbook link should hopefully help. The hard thing is finding the line between "this actually makes sense" and "I think this makes sense, when really it doesn't".

 

[QuantumCurt]

The hard thing is finding the line between "this actually makes sense" and "I think this makes sense, when really it doesn't".

Don't worry, I don't think I've found that line yet either.

 

[Physics-UG]

Don't worry, I don't think I've found that line yet either.

Haha. Well then I'm looking like I'm set!

Thanks for the help everyone, I appreciate it.

 

[MidgetDwarf]

Whay book are you guys using?

 

[Mark44]

Before the midterm I was doing the assignments and, for the most part, they made sense. I didn't really do textbook based problems because it felt like there was a huge leap between the 2 and the textbook was subpar when it came to it's usefulness.

Did the assignments come from some other source than the book?

In my experience of teaching linear algebra many times, I had a lot of students struggle with this class more than they would in the calculus courses. The reason is that linear algebra is less process-oriented than a lot of first-year calculus, and entails more problems where the student has to prove results, rather than plug stuff into a cookbook recipe. Some of the definitions are pretty subtle - for instance, how the terms "linearly dependent" and "linearly independent" are defined. To many students, the definitions appear to be the same, so they have a hard time showing that a given set of vectors is linearly independent.

Linear algebra books tend to have fewer pictures than calculus books, so might that be a reason you doubt your textbook's usefulness?

 

[symbolipoint]

Mark44 says this about Linear Algebra books:

Linear algebra books tend to have fewer pictures than calculus books, so might that be a reason you doubt your textbook's usefulness?

Why is this fewer pictures common? Are such illustrative or instructive pictures rare because of the nature of linear algebra?

 

[Intraverno]

Mark44 says this about Linear Algebra books:Why is this fewer pictures common? Are such illustrative or instructive pictures rare because of the nature of linear algebra?

Linear Algebra is very abstract, so that's probably the reason if I had to guess.

 

[MidgetDwarf]

Mark44 says this about Linear Algebra books:Why is this fewer pictures common? Are such illustrative or instructive pictures rare because of the nature of linear algebra?

I believe that it's hard to picture anything above 3rd dimension. I wish most linear algebra books had geometric intuition attached to it. I'm currently taking linear algebra. I sympathize with OP. We are using Anton Elementary Linear Algebra. The book is rather confusing to read, and the examples are ussually not worked out and only the results given. The book takes a nose dive and becomes the book of he'll in chapter 8 (general linear transformation s). Combined with an extremely bad instructor, the difficulty of linear algebra becomes graduate level (I'm exaggerating a bit).

The material is not hard, rather it depends largely how the instructor presents it. When a professor has groups of students ( 90 percent of the class in every one of the sections my teacher is teaching/from algebra to diff eq) who are failing year after year, it is time to have an instructor sit in his lecture for at least a month and kick the professor out. No matter how good of a researcher, accolades, or popularity among the faculty.

Best advices is to get a copy of linear algebra done right (Axler), to see how the theme develops and tie in together. Maybe Axler is beyond your understanding, or scope of class. Read it like a novel, ask so what, and just work as hard you can to understand.

 

[Physics-UG]

Whay book are you guys using?

We're using David Poole Linear Algebra, a modern introduction (this - http://www.amazon.com/dp/1285463242/?tag=pfamazon01-20 )

Did the assignments come from some other source than the book?

In my experience of teaching linear algebra many times, I had a lot of students struggle with this class more than they would in the calculus courses. The reason is that linear algebra is less process-oriented than a lot of first-year calculus, and entails more problems where the student has to prove results, rather than plug stuff into a cookbook recipe. Some of the definitions are pretty subtle - for instance, how the terms "linearly dependent" and "linearly independent" are defined. To many students, the definitions appear to be the same, so they have a hard time showing that a given set of vectors is linearly independent.

Linear algebra books tend to have fewer pictures than calculus books, so might that be a reason you doubt your textbook's usefulness?

The assignments come from a combination of my prof and the prof teaching the other section. So, I'm not sure if they make them up for what they do.
I definitely find calculus to be quite.. Not simple, but easy to grasp. Especially in comparison to linear algebra haha...

I believe that it's hard to picture anything above 3rd dimension. I wish most linear algebra books had geometric intuition attached to it. I'm currently taking linear algebra. I sympathize with OP. We are using Anton Elementary Linear Algebra. The book is rather confusing to read, and the examples are ussually not worked out and only the results given. The book takes a nose dive and becomes the book of he'll in chapter 8 (general linear transformation s). Combined with an extremely bad instructor, the difficulty of linear algebra becomes graduate level (I'm exaggerating a bit).

The material is not hard, rather it depends largely how the instructor presents it. When a professor has groups of students ( 90 percent of the class in every one of the sections my teacher is teaching/from algebra to diff eq) who are failing year after year, it is time to have an instructor sit in his lecture for at least a month and kick the professor out. No matter how good of a researcher, accolades, or popularity among the faculty.

Best advices is to get a copy of linear algebra done right (Axler), to see how the theme develops and tie in together. Maybe Axler is beyond your understanding, or scope of class. Read it like a novel, ask so what, and just work as hard you can to understand.

I find that picturing it isn't hard. But, I do find this text confusing to read as well. As said above we are using a different text than you, but I definitely don't know if I like it.. Someone I know said they love this text, so I'm going to pick their brain and see if they know something I don't.

I do feel the same way in terms of difficulty. It's not that it's difficult, there's just some portion or step that I seem to be missing to make that connection. Certain things make total sense, and others are a blur.

 

[MidgetDwarf]

We're using David Poole Linear Algebra, a modern introduction (this - http://www.amazon.com/dp/1285463242/?tag=pfamazon01-20 )
The assignments come from a combination of my prof and the prof teaching the other section. So, I'm not sure if they make them up for what they do.
I definitely find calculus to be quite.. Not simple, but easy to grasp. Especially in comparison to linear algebra haha...
I find that picturing it isn't hard. But, I do find this text confusing to read as well. As said above we are using a different text than you, but I definitely don't know if I like it.. Someone I know said they love this text, so I'm going to pick their brain and see if they know something I don't.

I do feel the same way in terms of difficulty. It's not that it's difficult, there's just some portion or step that I seem to be missing to make that connection. Certain things make total sense, and others are a blur.

My buddy took at another college using poole, and he mention to have had trouble with the book. Really smart guy, has worked through courant, hardy books, and he still quite did not understand that book.

 

[Physics-UG]

My buddy took at another college using poole, and he mention to have had trouble with the book. Really smart guy, has worked through courant, hardy books, and he still quite did not understand that book.

Oh, perfect! At least I know I'm not completely alone on this one haha.