Jon

Hey, I am new at real analysis and I have been practicing Calculus the limits, sequences series, and convergence tests. Overall, what is the best way to study for real Analysis? I am reading Real Analysis and Foundations by Steven Krantz. But I feel like I am reading it passively and not really understanding the proofs in the book. What is the best way to prepare for real analysis so that one can really understand the proofs. For example the proofs about the Number Systems.

Has anyone here ever taken real analysis before?

No

Thanks

Sure

In Linear algebra what does $M \circ C $ mean if M and C are linear operators?

@feynhat sorry to sound redundant but then is $L(x)$ V and $x$ is W right? In the graph I provided earlier.

@LeakyNun So just use the II, and III. I is already confirmed by II and III yes?

I think I fixed it

$\text{Hey I have a general question about linear transformations as long as I confirm that } L(v_1+v_2) = L(v_1)+L(v_2), \\ \text{ and } L(\alpha v) = \alpha L(v) \\ \text{do I need to show}$ $L(\alpha v_1 + \beta v_1) = L(\alpha v_1) + L(\beta v_2) \text{Or is this already inferred}?$

Never mind then I was thinking as an algorithm.

There is Simpsons double integral math.stackexchange.com/questions/2554573/… @Mikhail

I see so $x_2$ is the only one to change because the reflection is just on the $x_2$ axis not on the $x_1$ axis.

$x_2 $ axis.

I don't understand. L is the linear transformation?

Yeah wouldn't a reflection be the opposite like $(-x_1,-x_2)$?

So $(-1,1)$ and $(1,1)$ are the points

First point is $L(x) =(-x_1,x_2)$ second point $(x_1,x_2)$

I have question regarding the geometric effect of a linear transformation for example the linear transformation $R^2$ to $R^2$ which is $L(x) = (-x_1,x_2)^t$ my main question about this question is I know that if you say$x_1 =1 , x_2 = 1$ you can plot the first point but what about the second what exactly is a reflection? Why is the reflect $(x_1,x_2)^t$?

Yours

Its the same thing just better notation

I did to verify that the LHS = RHS. In the end thats how we know it is a linear transformation.

@Ted So to brief this is the correct way of solving it right.$L(x) = 3x \\ v_1 = x, v_2 =y \\ L(\alpha(v_1)) = L(\alpha(x)) \\ L(\alpha x) = 3 \alpha x \\ \alpha L(v_1) = \alpha(3x) = \alpha 3x \\ L(v_1+v_2) = L(v_1) + L(v_2) \\ L(x+y) = 3(x+y) = 3x+3y \\ L(x)+L(y)= 3x+3y$

good good

@AbhishekBhatia What industry are you looking to pursue?

I guess more like analyzing the time complexity in algorithms like insertion sort. I got one more year then I got my bachelors.

Math with cs concentration, and perhaps a master in a field of data science.

Background : just a student.

The goal

Data science

Scientific Computing

I have taught my self linear algebra first through a numerical analysis textbook then the Leon book is this the right way?

@Ted How do you get good at advanced Linear Algebra.

What is a standard book?

Linear Algebra With Applications 7th Edition by Steven J.Leon

mhhhhh.....

You can say $L(x) + L(y) = 3(x) + 3(y)$ so $v_1 = x, v_2 = y$?

$L(x+y)=3\cdot (x+y)$

I forgot to add that $v= (x_1,x_2)^T$ if that clears up any misunderstanding.

One last question thanks for your time what is $v_1$ and $v_2$ equate to?

Sorry then what is x then if not a vector?

And rightly so.

It aligns with the book therefore it must be right.

I know no other way to solve other than using the method I just mentioned.

Are you sure?

Its from $L:V to V$ to itself so $R^2$

Can I use math jax here?

No worries Ted answered my question