Linear transformation arbitrary question

[cal.queen92]

Homework Statement

Suppose A is an mxn matrix and b is a vector in R^m. Define a function T:R^n --> R^m by T(x) = Ax + b. Prove that if T is a linear transformation then b=0.

Homework Equations

For the second part of the question, a transformation is linear if:

1) T(u+v) = T(u) + T(v) for all u,v in domain of T

2) T(cu) = cT(u) for all u & c (scalars)

The Attempt at a Solution

For the first part of the question, I am thrown off by the "+b" as I never saw anything other than T(x) = Ax.

However, I am leaning towards using the identity matrix I to prove T(x) = Ax, but I don't feel this will be complete...

I really need a hand getting started and I think I'll be able to pick it up from there.

Thank you!

 

[micromass]

Calculate T(u+v) and T(u)+T(v) with your definition of T. Are those two equal?

 

[cal.queen92]

Thank you!

So I gave numerical values to the matrix A, & vectors b, u &v (making b a zero vector) and was able to conclude that T(u+v) = T(u) + T(v).

Since I made b a zero vector, this helps me with the second part of the problem. However, how can I connect this to the first part of the question?

Does proving this define the function? What does it mean to define the function?

 

[Dick]

Put u and v equal to the zero vector. So if T is linear, then T(0)+T(0) should equal T(0). If T(u)=Au+b, what does that tell you about b?

 

[cal.queen92]

Okay, I made u & v zero vectors so that T(u+v) = T(u) + T(v). (T is linear)

Now, one rule I have says that to define the function (or for T to be linear), T(0) = 0 (has to be so)

So this means that if T(u) = A(u) + b, then b has to be a zero vector as well right?

Does proving that a transformation T is linear help me in defining a function? What does it mean to define a function? (this may be a very basic question, but I can't grasp it!)

 

[Dick]

Okay, I made u & v zero vectors so that T(u+v) = T(u) + T(v). (T is linear)

Now, one rule I have says that to define the function (or for T to be linear), T(0) = 0 (has to be so)

So this means that if T(u) = A(u) + b, then b has to be a zero vector as well right?

Does proving that a transformation T is linear help me in defining a function? What does it mean to define a function? (this may be a very basic question, but I can't grasp it!)

Sure. If T is a linear function then T(0) has to equal 0. So b has to be 0. I'm a little vague on what the rest of your question is.

 

[cal.queen92]

You have helped a lot, thank you!

The rest of my question is really about the first part of the statement:

"Suppose A is an mxn matrix and b is a vector in R^m. Define a function T:R^n --> R^m) by T(x) = Ax + b."

Perhaps I'm over thinking it, but what does it mean to define a function in this way? How does this relate to proving that b=0 if the transformation is linear?

 

[Dick]

You have helped a lot, thank you!

The rest of my question is really about the first part of the statement:

"Suppose A is an mxn matrix and b is a vector in R^m. Define a function T:R^n --> R^m) by T(x) = Ax + b."

Perhaps I'm over thinking it, but what does it mean to define a function in this way? How does this relate to proving that b=0 if the transformation is linear?

Yeah, maybe overthinking it. A is a matrix with a bunch of numbers in it. No matter what those numbers are A operating on the zero vector is the zero vector. Any number times 0 is 0. So T(0)=b.

 

[cal.queen92]

hehe Okay, I understand, but...

Just one last question:

Does what we did here (proving that b = 0 makes the transformation linear) ALSO define the function as asked in the first part of the problem statement?

 

[micromass]

hehe Okay, I understand, but...

Just one last question:

Does what we did here (proving that b = 0 makes the transformation linear) ALSO define the function as asked in the first part of the problem statement?

I really don't understand. Nobody is asking you to define a function. There are no two parts in the problem statement.
The only thing they say is that you have a function f(x)=Ax+b for a matrix A and a column vector b.

 

[Dick]

hehe Okay, I understand, but...

Just one last question:

Does what we did here (proving that b = 0 makes the transformation linear) ALSO define the function as asked in the first part of the problem statement?

No, T(x)=Ax+b defines a function. For it to be linear means b is zero. That doesn't define A at all. Why do you think it should?

 

[cal.queen92]

I'm not sure... I think I'm really just having trouble figuring out with this question is asking.

From what I understand, together, we did the proof that was required (b=0). Is that all that is being asked for?

 

[cal.queen92]

Definitely over thinking this, think I just had a light bulb.

Thank you guys for your time and patience!