Linear Transformations

Which of the following statements is(are) TRUE?

• There does not exist a linear transformation T: R² →  R² such that T(2, 1) = (5, 6); T(1, -1) = (2, 3), T(4, 5) = (11, 12).
• There exists a linear transformation T: R³ →  R² such that T(1, 1, 1) = (1, 2); T(1, 1, 0) = (2, 1 ), T(1, 0, 0) = (1, 1); T(3, 1, 2) = (2, 6)
• There exists more than one linear transformations T: R² →  R²  such that T(2, 3) =(1, 0) and T(3, 2) =(-2, 3).
• A linear transformation T: R² →  R² is uniquely determined if T(1, 2) and T(3, 5) are known.

Let T: R³ →  R³ be a linear transformation such that T(a, b, c) = (a - c, b + 2c, a + b - c).

• The null space of T is xy- plane.
• The range space of T is z-axis.
• The rank of T is 3.
• T is singular.

Which of the following is NOT a linear transformation?

• T: R³ →  R³ given by T(x, y, z) = (9x + 4y - z, x + y - z, y + 3z).
• T: R² →  R² given by T(x, y) = (0, x).
• T: R³ →  R² given by T(x, y, z) = (x, yz, z).
• T: R³ →  R³ given by T(x, y, z) = (x -y, y - z, z -x).

The determinant of the matrix representation of T: R³ →  R³ given by T(x, y, z) = (x + 2y, y + z, 2x) with respect to the ordered bases {(1, 1, 1), (1, 1, 0), (1, 0, 0)} and {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is

• 4
• 8
• - 4
• - 8

Pick the correct statement.

• For any linear transformation T: V →  W, the relation:  Rank(T) + Nullity(T) = dim V holds.
• There exists a linear transformation T: V →  W which is one-to-one but not onto.
• If V is finite dimensional, then there is NO one-to-one linear transformation T: V →  V which is NOT onto.
• If a linear transformation T: V →  W is one-to-one, then it is necessarily invertible.