Question

Let t be a linear transformation from R² to R² such that [math]t(1,0) = (2,3)[/math] and [math]t(0,1) = (4,5)[/math]. Find [math]t(3,2)[/math].

Answer 1

To find t(3,2), we use the linearity of the transformation t, expressing (3,2) as a linear combination of the basis vectors (1,0) and (0,1), whose images under t are (2,3) and (4,5) respectively. The resulting transformation is t(3,2) = (14,19).

Step-by-step explanation:

To find the image of the vector (3,2) under the linear transformation t, we can use the linearity property of transformations. This property tells us that the transformation of a linear combination of vectors is the same as the linear combination of the transformations of those vectors.

Since we know t(1,0) = (2,3) and t(0,1) = (4,5), we can express (3,2) as a linear combination of (1,0) and (0,1):

(3,2) = 3*(1,0) + 2*(0,1).

Now we apply the transformation t to this linear combination:

t(3,2) = t(3*(1,0) + 2*(0,1)) = 3*t(1,0) + 2*t(0,1) = 3*(2,3) + 2*(4,5) = (6,9) + (8,10) = (14,19).

Therefore, the image of the vector (3,2) under the transformation t is (14,19).