0-/ potnm JBTAIL MALNAGANI HOTE ASA TUUKeCNEA FC DAROTHEE Define the linear transformation Find the kernel T. (If there are an infinite number of solutions, use parameterization) Find the range of T. Find the nullity of T. Find the rank of T. Need Help?
The Correct Answer and Explanation is:
1. Kernel of T:
The kernel of a linear transformation T: V → W is the set of vectors in the domain V that are mapped to the zero vector in the codomain W. In other words, it’s the set of all vectors v for which T(v) = 0.
For example, if T is represented by a matrix A, the kernel is the solution to the homogeneous equation A * v = 0.
If there’s an infinite number of solutions, you can parameterize the solutions. Typically, you’d express the solutions in terms of free variables.
2. Range of T:
The range of T is the set of all possible outputs, i.e., the image of the transformation. If T is represented by a matrix A, the range is spanned by the columns of A.
In other words, the range consists of all vectors in the codomain W that can be written as T(v) for some vector v in the domain V.
3. Nullity of T:
The nullity of T is the dimension of the kernel of T. This is the number of free variables in the solution to A * v = 0. If the kernel has an infinite number of solutions, the nullity is the number of free variables (i.e., the dimension of the null space).
4. Rank of T:
The rank of T is the dimension of the range of T, or the number of linearly independent vectors in the image of T. If T is represented by a matrix A, the rank is the number of pivot columns in the reduced row echelon form of A.
General Steps for Solving:
- Write down the transformation matrix A (if available).
- Solve A * v = 0 for the kernel.
- Find the rank (number of pivot columns) of A for the range and rank.
- The rank-nullity theorem states that: rank(T)+nullity(T)=dimension of domain.\text{rank}(T) + \text{nullity}(T) = \text{dimension of domain}.rank(T)+nullity(T)=dimension of domain.
Once I have the form of the transformation T or the matrix, I can give you a more specific solution with detailed steps!
