Question

Find a basis for the range and nullspace of the following linear mapping and verify the Rank-Nullity Theorem II b = a a+b atc a+b+ L : Rj → M(2.2) defined by L

Answer 1

To find a basis for the range and nullspace of the linear mapping defined by L(a, b, c) = (a, a + b, a + b + c), we need to determine the vectors that form a basis for the range and nullspace.

Step-by-step explanation:

To find a basis for the range and nullspace of the linear mapping defined by L(a, b, c) = (a, a + b, a + b + c) in R^3 to M(2,2), we need to determine the vectors that form a basis for the range and nullspace.

The range of L is the set of vectors in the codomain M(2,2) that are reached by applying L to vectors in the domain R^3. Since the codomain M(2,2) has dimension 4, we need to find a set of linearly independent vectors in M(2,2) that are in the range of L.

The nullspace of L is the set of vectors in the domain R^3 that are mapped to the zero vector in M(2,2). To find the nullspace, we can solve the equation L(a, b, c) = (0, 0, 0) and determine the conditions on a, b, and c that satisfy this equation.