Question

Properties of Linear Transformations: A transformation T: Rn > Rm is linear if and only if the following relationships hold for all vectors u and v in Rn and for every scalar c.
(a) T(u+v) = T(u) + T(v)
(B) T(cu) = cT(u)

Anyone want to explain what this means and then apply the theorem to T(x, y) = (2x + y, x-y) to see if the transformation is a linear operator?

Answer 1

I just watched a video of this on khanacademy.org, its called "Matrix Vector Products for Linear Transformations". Check it out - he explains everything.

Answer 2

(a) and (b) just prove that every matrix vector product can be used to find linear transformations. I'm stumped on how to solve your problem, though :(

Answer 3

I get that we have to plug in the u and v for the right side but how do you apply T(u+v)

Answer 4

I'm not sure, honestly, but check out that video I just mentioned, he explains it step by step.

Answer 5

im posting a solution that shows T(u+v)=T(u)+T(v), it should be clear how to take care of part b after that.

Answer 6

Its just a little bit of algebraic regrouping.

Answer 7

Ohhhh, I get it. Thanks joe!

I swear I have the absolute worst textbook in the history of textbooks. They're stingy on examples and love to jump all over the place.

Answer 8

I recommend reading Gilbert Strang's Linear Algebra and its Applications. My undergrad Linear Algebra course taught me nothing, and after spending a summer reading that book, now im taking graduate lvl linear algebra courses as an undergrad. it makes everything really clear.

Answer 9

I'll look into it. Thanks.

Linear algebra has some pretty funky abstract stuff. I liked calc a lot more.

Answer 10

im the complete opposite. I love Linear/Abstract Algebra/Topology/etc, and dislike Calc/Differential Equations/Real Analysis/etc.

Answer 11

What major are you pursuing?

Answer 12

Pure Math. Looking into doing research. what about you?

Answer 13

Still a high school student, but looking into applied math. Pure math sounds pretty difficult lol. Where are you studying?

Answer 14

Im at the University of Texas in San Antonio (not Austin which is the one most people know about). Pure Math is a little rough, but i was really never good with numbers, so the abstract math suits me just fine :)

Linear Algebra in high school, i wish i was that ambitious when i was in high school >.< keep going!

Answer 15

Yeah.. haha.. school budget problems and terrible scheduling left me without a science class this year so I decided to fill up the gap with an extra math class... AP stats at my school is a joke; it's more reading comprehension than actual math.

the abstract math is what i'm afraid of lol. i love the number crunching.