Finding the area/volume for a transformed parallelogram/parallelepiped?

[richashah]

Homework Statement

for the following, find (a) the image of the indicated set under the given linear transformation and (b) the area or volume of the image

question 1: P is the paralellogram in R^2 with corner (0,0) spanned by (1,-1) and (4,7);
It undergoes the transformation T(x) = (3x + 4y, 4x + 5y)

question 2: P is the parallelepiped with corner (1,1,1) spanned by (1,1,2), (1,2,1), and (2,1,1). T(x) = (3y, -4x, 5z)

Homework Equations

I know the area of a parallelogram = ||a x b|| (cross product between vectors a and b)
The volume of a parallelepipid = ||a x b|| . c (cross product of vectors a & b. Then the dot product is used between that answer and vector c to obtain volume)

The area of a transformed parallelogram = |det(transformation matrix)|*(area of parallelogram)
The volume of a transformed parallelepiped = |det(transformation matrix)|*(volume of parallelepipid)

The Attempt at a Solution

for question 1, I have the transformation points as (0,0) (-1,-1) and (-16,-19). For the area, I took the cross product of vectors a and b (vector a = (-1,-1) vector b = (-16,-19)) and got 3. I got the determinant equals 1. So i got the area to be 3, but the answer key says it is 11.

For question 2, I'm just completely confused to be quite honest.

 

[LCKurtz]

Homework Statement

for the following, find (a) the image of the indicated set under the given linear transformation and (b) the area or volume of the image

question 1: P is the paralellogram in R^2 with corner (0,0) spanned by (1,-1) and (4,7);
It undergoes the transformation T(x) = (3x + 4y, 4x + 5y)

question 2: P is the parallelepiped with corner (1,1,1) spanned by (1,1,2), (1,2,1), and (2,1,1). T(x) = (3y, -4x, 5z)

Homework Equations

I know the area of a parallelogram = ||a x b|| (cross product between vectors a and b)
The volume of a parallelepipid = ||a x b|| . c (cross product of vectors a & b. Then the dot product is used between that answer and vector c to obtain volume)

The area of a transformed parallelogram = |det(transformation matrix)|*(area of parallelogram)
The volume of a transformed parallelepiped = |det(transformation matrix)|*(volume of parallelepipid)

The Attempt at a Solution

for question 1, I have the transformation points as (0,0) (-1,-1) and (-16,-19). For the area, I took the cross product of vectors a and b (vector a = (-1,-1) vector b = (-16,-19)) and got 3. I got the determinant equals 1. So i got the area to be 3, but the answer key says it is 11.

For question 2, I'm just completely confused to be quite honest.

You need to show what you did. For (a) Show how you got the area of the of the parallelogram and the determinant of the transformation. That's all you need to calculate the new area. I think you have arithmetic mistakes.