Question

If a 3×4 matrix has rank 3 , what are the dimensions of its columnspace (e.g., which of R1,R2,…Rn represents the column space) and left nullspace (i.e., for a matrix Am×n, the left null space is the set of all vectors x such that A^T x=0) ?

Answer 1

The dimensions of the columnspace of a 3x4 matrix with rank 3 are 3x1 or R^3, and the dimensions of its left nullspace are 1x1 or R^1.

Step-by-step explanation:

The dimensions of the columnspace of a matrix are determined by the number of linearly independent columns in the matrix. If a 3x4 matrix has rank 3, it means that there are three linearly independent columns in the matrix. Therefore, the dimensions of its columnspace would be 3x1 or R^3.

The dimensions of the left nullspace of a matrix are determined by the number of linearly independent rows in the matrix. In this case, since the matrix is a 3x4 matrix, it would have 4-3=1 dependent row. Therefore, the dimensions of its left nullspace would be 1x1 or R^1.

Answer 2

The column space of the 3x4 matrix has a dimension of 3, and the left nullspace has a dimension of 1.

Step-by-step explanation:

The column space of a matrix represents the span of its column vectors.

In this case, since the rank of the matrix is 3, it means that there are 3 linearly independent columns in the matrix.

Therefore, the dimension of the column space is 3.

The left nullspace of a matrix is the set of all vectors that, when multiplied by the transpose of the matrix, result in the zero vector.

In this case, since the matrix is a 3x4 matrix, the transpose will be a 4x3 matrix.

The dimension of the left nullspace is then 4-3=1.