Question

Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2, ... are not vectors but are entries in vectors.

T(x1, x2, x3)= (x1-5x2=4x3, x2-6x3)

Answer 1

To show that T is a linear transformation, we need to prove that it satisfies two properties: additivity and homogeneity.

Step-by-step explanation:

To show that T is a linear transformation, we need to prove that it satisfies two properties: additivity and homogeneity.

1. Additivity: Let's take two vectors x and y and their corresponding entries x1, x2, and x3, y1, y2, and y3, respectively. We can calculate T(x) and T(y) as follows:

• T(x) = (x1 - 5x2 + 4x3, x2 - 6x3)
• T(y) = (y1 - 5y2 + 4y3, y2 - 6y3)

If we add T(x) and T(y):

T(x) + T(y) = (x1 - 5x2 + 4x3 + y1 - 5y2 + 4y3, x2 - 6x3 + y2 - 6y3)

Now, let's calculate T(x + y):

T(x + y) = ( (x1 + y1) - 5(x2 + y2) + 4(x3 + y3), (x2 + y2) - 6(x3 + y3) )

We can see that T(x) + T(y) = T(x + y), which satisfies the property of additivity.

2. Homogeneity: Let's consider multiplying a vector x by a scalar c. We can calculate T(x) and T(c * x) as follows:

• T(x) = (x1 - 5x2 + 4x3, x2 - 6x3)
• T(c * x) = (c * x1 - 5c * x2 + 4c * x3, c * x2 - 6c * x3)

We can see that each entry in T(c * x) is c times the corresponding entry in T(x). This satisfies the property of homogeneity.

Therefore, T is a linear transformation.