Final answer:
A 3×4 matrix with rank 3 has a three-dimensional column space that spans ℝ³ and a trivial left nullspace containing only the zero vector.
Step-by-step explanation:
If a 3×4 matrix has rank 3, this indicates that the matrix has three linearly independent rows and therefore three linearly independent columns. The column space of a matrix is defined as the span of its columns. Since the matrix has rank 3, all three of its columns contribute to the column space, which means the column space is three-dimensional and spans ℝ³, where ℝ denotes the set of real numbers.
On the other hand, the left nullspace (also known as the kernel or null space of the transpose of the matrix) consists of all row vectors that, when left-multiplied with the matrix, result in the zero vector. For a matrix of rank 3, the dimension of the left nullspace is given by the number of rows minus the rank, which in this case is 3-3=0. Therefore, the left nullspace only contains the zero vector, meaning it is trivial.