chick the attachments from question 3 to 7

as you see I have attached two attachments of the questions that i need their answers I just need question 3,4,5,6,and 7 to be answered thanks
1
Let A =
-2] and define T: R? – R by T(x) = Ax. Find the images under T of u = |
=[]
= 6.5), and the vector (u + v). Also sketch the three vectors before and after the
transformation. Write 1-2 sentences that explain how you found your answers/knew they were
right and why the sketch makes sense.
VE
–5
2. Let A = 5] and define T:R→ Rby T(x) = Ax.
7
Find the image under T of u =
b. Find a vector x whose image under T is b = [ 12). Explain why your work makes
-0.)
а.
sense.
3. [If not done in class] Suppose that a transformation T: R2 → R2 is defined by the matrix
A = [a b]. In order to do this, you must show that the definition of linear transformation is
satisfied with this matrix A. To help you get started, let x = 6;] and y = P:]
number. You need to algebraically show both that A(x + y) = Ax + Ay and A(cx) = cA(x).
and cbe a real
(x, +
4. Show that the transformation T defined by T
is not a linear transformation.
5. TRUE OR FALSE: (Assume that the product AB is defined). If the columns of B are linearly
dependent, then so are the columns of AB. If TRUE, provide a justification. If FALSE, provide a
counterexample.
6. After class, two linear algebra students start talking about linear transformations and the letter
“N.” One of the students suggested translation (shifting up) as another linear transformation that
could be done to the letter “N,” like
N.
N
mot
(003
The other student disagreed, stating that shifting the “N” up like this is NOT an example of a
linear transformation. Which student is right? Why?
7. Consider the image given below and the transformation matrix C =
=10 15
с
NH
a.
Sketch what will happen to the image under the transformation.
b. Describe in words what will happen to the image under the transformation.
C. Describe how you determined that happened. (What, if any, calculations did
you do? Did you make a prediction? How did you know you were right? etc.)