Suppose T: ℝ3→ℝ2 is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(U+2V).
U = [1, -2, -1]^T
V = [2, -2, -5]^T
T(U) = [-1, 8]^T
T(V) = [3, 13]^T
To find T(U+2V), we need to determine the transformation of the vector U+2V.
First, let's find the value of U+2V.
U + 2V = [1, -2, -1] + 2[2, -2, -5]
= [1, -2, -1] + [4, -4, -10]
= [1+4, -2-4, -1-10]
= [5, -6, -11]
Now, we need to find T(U+2V) by applying the linear transformation T to the vector [5, -6, -11].
Since T is a linear transformation from ℝ3 to ℝ2, it means that T maps a 3-dimensional vector to a 2-dimensional vector.
To determine T(U+2V), we can evaluate it as follows:
T(U+2V) = T([5, -6, -11])
To find T([5, -6, -11]), we can use the transformation properties of T:
T([5, -6, -11]) = T(5[1, 0, 0] - 6[0, 1, 0] - 11[0, 0, 1])
Since T is a linear transformation, we know that T(aU + bV) = aT(U) + bT(V), where a and b are scalars, and U and V are vectors.
Using this property, we can rewrite T([5, -6, -11]) as:
T(5[1, 0, 0] - 6[0, 1, 0] - 11[0, 0, 1]) = 5T([1, 0, 0]) - 6T([0, 1, 0]) - 11T([0, 0, 1])
Since T(U) and T(V) are given, we can substitute their values:
T([5, -6, -11]) = 5T([1, 0, 0]) - 6T([0, 1, 0]) - 11T([0, 0, 1])
= 5[-1, 8] - 6[3, 13] - 11[0, 0]
Now, compute the multiplication:
= [-5, 40] - [18, 78] - [0, 0]
= [-5-18-0, 40-78-0]
= [-23, -38]
Therefore, T(U+2V) = [-23, -38]