Question

For the following transformation T either
give its standard matrix (i.e. the matrix relative to the standard bases of the domain of T and the codomain of T), if T is linear, or
find a counter-example that demonstrates T is not linear. (see question for full T)... help, please? :S

Answer 1

T\[{ \left(\begin{matrix}x_1 \\ x_2\\\ x_3\\\ x_4\end{matrix}\right) }\] = \[\left(\begin{matrix}x_1 - 5 x_2 - 6x_3 + x_4 \\ 2x_2 - x_3 - x_4x_3\end{matrix}\right)\]

(sorry about the formatting)

Answer 2

is \[T\left[2{ \left(\begin{matrix}1 \\ 1\\\ 1\\\ 1\end{matrix}\right) }\right]=2T\left[{ \left(\begin{matrix}1 \\ 1\\\ 1\\\ 1\end{matrix}\right) }\right]\]

Answer 3

... Yes?

Answer 4

no

Answer 5

oh :(

Answer 6

haha

Answer 7

wait - ill give that an actual go, rather than just guessing...

Answer 8

ok ;)

Answer 9

OK, so the first (T(2[x_1 ... ])) is [-18, -2], and the other is [-22,0] ... or something..? :/

Answer 10

-22 in not correct

Answer 11

is not

Answer 12

it should be -18

Answer 13

the -2 and 0 are correct...and that makes it not linear

Answer 14

\[x_4\cdot x_3\] is a problem

Answer 15

In general for a transformation to fail to be linear it must fail to satisfy one of these two properties:

\[T(u + v) = T(u) + T(v)\]\[T(av) = aT(v)\]

Answer 16

As Zarkon demonstrated \(T(av) \neq aT(v)\)

Answer 17

Cheers - don't know why I struggle so much with this stuff!! I'm a calculus major, and have done 2 years of engineering mathematics, but linear algebra just does my head in! Expect a few more questions as the semester goes on haha thanks!

Answer 18

"calculus major"

didn't know there was such a thing

Answer 19

In Australia - might be different to wherever you are :)

Answer 20

I'm in the USA