Question

Consider a transformation T: R^3 -- R^2 defined by
T( x1,x2,x3)= (2x1-3x2+x3, 4x3-x1)

(A) show that the tranformation, T, is linear.

(b) Find T( e1, e2,e3) using these, determine the standard matrix of the transformation.

(c) find the image of v=(2, 0, 1 ) under the linear mapping T as a linear comination of T(e1,e2,e3)

(d) is T one-to-one? Does Ta map R^3 onto R^2? Explain

Answer 1

(A) Take\[(x_1,x_2,x_3)\] and \[(x_1',x_2',x_3')\] as two set of vectors.
Do the transformation.
It would be equal to T\[(x_1,x_2,x_3)\]+T\[(x_1',x_2',x_3')\].

(B) Take the basis vector and find its corresponding transform.
The transformed vector thus obtained forms the column of T.

(C) Ans=T.v

(D) Yes, matrix transformation is one to one. T maps \[\mathbb{R}^3\] to \[\mathbb{R}^2\] which is evident from the fact that size of T matrix is 2*3.