Final answer:
The question involves determining the Linear Transformation from №³ (ℝ) to №² (ℝ) using the given matrix [ 1 -1 2 3 1 0 ] and the standard basis. The transformation matrix is applied to the standard basis vectors to obtain their images in the two-dimensional space, resulting in the vectors [1, 3], [-1, 1], and [2, 0].
Step-by-step explanation:
The question asks us to determine the Linear Transformation for a given matrix in relation to the standard basis for a transformation from №³ (ℝ) to №² (ℝ). This implies that we are looking at a transformation from a three-dimensional space to a two-dimensional space. The matrix provided is [ 1 -1 2 3 1 0 ], which suggests a transformation matrix that acts on vectors in №³ (ℝ).
The standard basis for №³ (ℝ) is e1 = [1, 0, 0], e2 = [0, 1, 0], e3 = [0, 0, 1]. Applying the linear transformation defined by the matrix to each of these basis vectors, we get the images in №² (ℝ) which would be two-dimensional vectors. The transformed vectors will be the columns of the resulting matrix, which represent the images of the standard basis vectors after the transformation.
For the given problem, we can represent the transformation matrix as:
T = [1 -1 2 ]
[3 1 0 ]
Applying T to e1 gives us the vector [1, 3], to e2 gives [ -1, 1], and to e3 gives [2, 0]. The images of the standard basis vectors under the linear transformation defined by T would be these resultant vectors.
Therefore, the Linear Transformation of №³ (ℝ) to №² (ℝ) with respect to the standard basis using the matrix [ 1 -1 2 3 1 0 ] is a function that maps e1 to [1, 3], e2 to [ -1, 1], and e3 to [2, 0]. These images form the columns of the matrix that represents the transformation with respect to the standard bases of №³ (ℝ) and №² (ℝ), respectively.