Question

let p be the parallelogram determined by the vectors [4;1] and [3;-1]. let q be the shape obtained by applying the linear transformation t(x) = [3 1;1 2]x to the parallelogram p. fing the area of q. show all of your work.

Answer 1

To find the area of shape q, apply the linear transformation to the given parallelogram p and calculate the area using the magnitude of the cross product of the image vectors.

Step-by-step explanation:

To find the area of shape q, we need to apply the linear transformation t(x) = [3 1;1 2]x to the parallelogram p. Let's first calculate the image of the two given vectors under this transformation:

[3 1;1 2][4;1] = [15;6]

[3 1;1 2][3;-1] = [8;0]

Now, we can construct the parallelogram determined by these two image vectors. The area of q will be the same as the area of this parallelogram. Using the formula for the area of a parallelogram, which is given by the magnitude of the cross product of two side vectors, we can calculate the area as:

|[15;6] x [8;0]| = |(15)(0) - (6)(8)| = 48

Therefore, the area of shape q is 48 square units.

Answer 2

To find the area of shape Q after a linear transformation, we first apply the transformation to the vectors defining parallelogram P, and then calculate the determinant of the matrix formed by the transformed vectors. The absolute value of the determinant gives us the area of the new parallelogram.

Step-by-step explanation:

The question asks us to find the area of shape Q which is obtained by applying the linear transformation T(x) = [3 1; 1 2]x to the parallelogram P determined by the vectors [4; 1] and [3; -1]. To solve this, we apply the transformation T to each vector and then use the resultant vectors to find the area of the new parallelogram Q.

First, we apply the transformation T to the vectors that define parallelogram P:

• For vector [4; 1]: T([4; 1]) = [3 1; 1 2] * [4; 1] = [12+1; 4+2] = [13; 6]
• For vector [3; -1]: T([3; -1]) = [3 1; 1 2] * [3; -1] = [9-1; 3-2] = [8; 1]

The area of the new parallelogram Q can be found by the determinant of the matrix formed by the transformed vectors:

Area of Q = |13 8|
| 6 1| = (13*1 - 8*6) = 13 - 48 = -35

The area is the absolute value of the determinant, so the area of Q is 35 square units.