Let T : R3 →R3 be the linear transformation defined by T(x, y, z) = (x + y, y + z, z + x) for all (x, y, z) ∈ R3. Thena)rank(T) = 0, nullity(T) = 3b)rank(T) = 2, nullity(T) = 1c)rank(T) = 1, nullity(T) = 2d)rank(T) = 3, nullity(T) = 0Correct answer is option 'D'. Can you explain this answer?

Question

Let T : R3 &rarr;R3 be the linear transformation defined by T(x, y, z) = (x + y, y + z, z + x) for all (x, y, z) &isin; R3. Thena)rank(T) = 0, nullity(T) = 3b)rank(T) = 2, nullity(T) = 1c)rank(T) = 1, nullity(T) = 2d)rank(T) = 3, nullity(T) = 0Correct answer is option 'D'. Can you explain this answer?

Answer

Solution:
To find the rank and nullity of the linear transformation T, we need to find the kernel and image of T.

Finding Kernel of T:

Let (x, y, z) be a vector in the kernel of T.

Then, T(x, y, z) = (0, 0, 0)

Therefore, we have the following system of equations:

x + y = 0
z + z = 0
z + x = 0

Solving the above system of equations, we get:

x = -y
z = 0

Therefore, the kernel of T is spanned by the vector (-1, 1, 0).

Finding Image of T:

Let (a, b, c) be a vector in the image of T.

Then, there exists a vector (x, y, z) such that T(x, y, z) = (a, b, c).

Therefore, we have the following system of equations:

x + y = a
z + z = b
z + x = c

Solving the above system of equations, we get:

x = (a - b + c)/2
y = (b - a - c)/2
z = (b - c)/2

Therefore, the image of T is the set of all vectors of the form (a, b, c), where a - b + c = 0.

Finding Rank and Nullity of T:

The rank of T is the dimension of the image of T, which is 2.

The nullity of T is the dimension of the kernel of T, which is 1.

Hence, the correct option is (D) rank(T) = 3, nullity(T) = 0.

Answer

rank(T) = 2, nullity(T) = 1. The rank of a linear transformation is the dimension of its range, which in this case is 2. The nullity of a linear transformation is the dimension of its kernel, which in this case is 1.

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