Question

is the following linear transformation linear or not? if it is, what is matrix representation of the linear transformation? t(x, y, z) = (x 2 y, y − z)

Answer 1

The transformation t(x, y, z) = (x + 2y, y - z) is a linear transformation. Its matrix representation is obtained by applying the transformation to the standard basis vectors of R^3, resulting in the matrix [1 2 0; 0 1 -1].

Step-by-step explanation:

The given transformation t(x, y, z) = (x + 2y, y - z) is indeed a linear transformation because it satisfies the two properties required for a transformation to be linear:

1. Additivity: t(u + v) = t(u) + t(v) for any vectors u, v in the domain.
2. Scalar multiplication: t(cu) = ct(u) for any vector u in the domain and scalar c.

The matrix representation of the linear transformation can be found by transforming the standard basis vectors of the domain space and ordering the resultant vectors as columns in a matrix. For a transformation from R^3 to R^2, the matrix will have the form:

[t(e1) | t(e2) | t(e3)]

Where e1, e2, e3 are the standard basis vectors (1,0,0), (0,1,0), and (0,0,1) respectively. Therefore, applying the transformation t to these vectors gives us:

1. t(e1) = t(1, 0, 0) = (1, 0)
2. t(e2) = t(0, 1, 0) = (0 + 2*1, 1 - 0) = (2, 1)
3. t(e3) = t(0, 0, 1) = (0 + 2*0, 0 - 1) = (0, -1)

The resulting matrix is:

[
1 0 0
2 1 -1
]