Question

Let ( F: ᴿ³+ᴿ³ ) be a linear transformation such that ( T(1,1,1)=(4,0,-1), 7(0,-1,21=(-4,5,-1), and ( (1,0,1)=(1,0,1)). T(2,-1,-1) T(2,-3)

Answer 1

The question inquires about the image of a vector under a specified linear transformation, which is a topic in mathematics usually addressed at the college level.

Step-by-step explanation:

The question involves determining the result of a linear transformation in mathematics, specifically within the field of linear algebra. In the given scenarios, various vectors in ℝ³ are being transformed, and the aim is to find the image of a vector under the transformation described by the provided examples.

To tackle this question, one typically starts by expressing the standard basis vectors of ℝ³ as linear combinations of the given transformed vectors. The resulting system of equations would be solved to find the transformation matrix that characterizes the linear transformation. With the matrix established, the image of any vector under the transformation can be computed by matrix-vector multiplication.

Answer 2

The result of the linear transformation T on the vector (2, -1, -1) is (12, 5, -3), and for the vector (2, -3, 0), it is (-10, 15, -7).

Step-by-step explanation:

Given the linear transformation T: ℝ³ → ℝ³ with specific mappings, we can find the images of vectors (2, -1, -1) and (2, -3, 0). The transformation T is uniquely determined by its action on the standard basis vectors. Utilizing the provided mappings:

`\[T₁(1, 1, 1) = (4, 0, -1),\]`

`\[T₂(0, -1, 21) = (-4, 5, -1),\]`

`\[T₃(1, 0, 1) = (1, 0, 1).\]`

We can express vectors (2, -1, -1) and (2, -3, 0) as linear combinations of these basis vectors and use linearity to find their images under T. For (2, -1, -1), the image T(2, -1, -1) is obtained by adding twice the first mapping, subtracting once the second mapping, and subtracting once the third mapping. Similarly, for (2, -3, 0), the image T(2, -3, 0) is determined through twice the first mapping, three times the second mapping, and zero times the third mapping.

The calculated results are T(2, -1, -1) = (12, 5, -3) and T(2, -3, 0) = (-10, 15, -7). These represent the linear transformation T applied to the given vectors, providing a clear understanding of the images of these vectors under the specified linear transformation.