- #1

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Let L:R(4) goes to R(4) be the linear transformation defined by

-matlab notation, the value is a 4x1 column

L ( [ a b c d])=[ a-b

0

c-d

0 ]

Show directly L is linear.

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- Thread starter jlucas134
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- #1

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Let L:R(4) goes to R(4) be the linear transformation defined by

-matlab notation, the value is a 4x1 column

L ( [ a b c d])=[ a-b

0

c-d

0 ]

Show directly L is linear.

- #2

radou

Homework Helper

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In order to get help, you should show some efforts.

What is the definition of a linear operator?

What is the definition of a linear operator?

- #3

HallsofIvy

Science Advisor

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Write down the requirements for a linear transformation, insert your L, and do the calculations!

- #4

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I am still trying to figure out this message board...

I forgot to include what I already know...

I know you have to prove that L(u+v)=L(u)+ L(v) and L(c*u)=c*L(u), but I don't understand how to set it up.

I tried to separate it into a1 and a2, but just get confused...

Do I have to place it in the standard matrix representation then solve?

- #5

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still working with it...

if I set u=a1, b1, c1, d1 and v=a1, b1, c1, d1

set L(u+v)=(this is what I get)

[ (a1+a2)-(b1+b2)

0

(c1+c2)-(d1+d2)

0 ]

which equals L(u)+L(v)

for L(c*u)=c*L(u)

[c(a-b) c*0 c(c-d) c*0]

which converts to c[a-b 0 c-d 0] which breaks down to c*L(u)

Is the close?

if I set u=a1, b1, c1, d1 and v=a1, b1, c1, d1

set L(u+v)=(this is what I get)

[ (a1+a2)-(b1+b2)

0

(c1+c2)-(d1+d2)

0 ]

which equals L(u)+L(v)

for L(c*u)=c*L(u)

[c(a-b) c*0 c(c-d) c*0]

which converts to c[a-b 0 c-d 0] which breaks down to c*L(u)

Is the close?

Last edited:

- #6

HallsofIvy

Science Advisor

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(except that you said " I set u=a1, b1, c1, d1 and v=a1, b1, c1, d1" when you mean "I set u=a1, b1, c1, d1 and v=a2, b2, c2, d2". Doesn't your class or textbook have some convention for writing vectors- say (a, b, c, d) or <a, b, c, d> rather than just a, b, c, d which can be confusing?

- #7

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When I first did it, it didn't look right...almost too simple to be correct. Thanks for correcting me with my notation.

my text does have a format but this doesn't support the large brackets required for a 4X1 matrix.

I will try to make the matrix a little more easier to read...

Would the notation for MATLAB entry suffice?

- #8

radou

Homework Helper

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my text does have a format but this doesn't support the large brackets required for a 4X1 matrix.

You can always write (a b c d)^T to represent a 4x1 matrix of you want.

- #9

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Can someone verify my signs are correct for when I solved, with help from the board?

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