Question

Let e₁ - [1] , e₂ = [0], y₁ =[2], and y₂ = [-1]

[0] [1] [5] [6]
and let T : R2 → R2 be a linear transformation that maps e1 into y1 and maps e2. find the images of
[5 ] and [x₁]
[-3] [x₂]

Answer 1

To find the images of [5, -3] and [x1, x2] under the linear transformation T, we express them as linear combinations of basis vectors e1 and e2 and apply T to each to get [7, -10] and [2x1 - x2, x1 + 5x2] respectively.

Step-by-step explanation:

The student is asking to find the images of vectors under a given linear transformation T from R^2 to R^2, where T maps e1 to y1 and e2 to y2. Using the assumption that T is a linear transformation, we can use the principle that T(ae1 + be2) = aT(e1) + bT(e2) for any vectors ae1 and be2 in R^2. The vectors in question can be expressed as a linear combination of the basis vectors e1 and e2.

For the vector [5, -3], we can represent it as a linear combination of e1 and e2, which is 5e1 - 3e2. Applying T, we get:

T([5, -3]) = 5T(e1) - 3T(e2) = 5[2, 1] - 3[-1, 5] = [5(2) + 3(1), 5(1) - 3(5)] = [10 - 3, 5 - 15] = [7, -10].

Similarly, for any vector [x1, x2], we have:

T([x1, x2]) = x1T(e1) + x2T(e2) = x1[2, 1] + x2[-1, 5] = [2x1 - x2, x1 + 5x2].