Question

suppose a 4 x 7 matrix a has three pivot columns. is col a = r^3? what is th edimension of nul a? explain your answers.

Answer 1

The column space of a 4 x 7 matrix A with three pivot columns can be represented as ℝ^3, and the dimension of Nul A is determined to be 4 by using the rank-nullity theorem.

Step-by-step explanation:

If a 4 x 7 matrix A has three pivot columns, this means that the rank of matrix A is 3. The column space of A, denoted as Col A, is the subspace of ℝ^4 spanned by the pivot columns. Since there are three pivot columns, the dimension of Col A, also known as the rank, is 3, which can be represented as ℝ^3 in terms of dimensionality. Therefore, Col A ≅ ℝ^3.

Regarding the dimension of the null space of A, also known as Nul A, the rank-nullity theorem can be applied. The theorem states that the dimension of the column space plus the dimension of the null space equals the number of columns in the matrix. In this case, we have:

• Dimension of Col A (rank) = 3
• Number of columns in A = 7

So, the dimension of Nul A = Number of columns - Rank = 7 - 3 = 4. Therefore, the dimension of Nul A is 4.

Answer 2

The column space of a 4 x 7 matrix A with three pivot columns is a subspace of ℝ^3, and the dimension of the null space of A is 4, determined by subtracting the rank from the number of columns.

Step-by-step explanation:

If a 4 x 7 matrix A has three pivot columns, this means that the rank of matrix A is 3. The column space of A, denoted as Col A, is spanned by the pivot columns and hence has a dimension equal to the rank of the matrix, which is 3. Therefore, Col A is a subspace of ℝ^3, the three-dimensional real coordinate space.

The dimension of the null space (Nul A), also known as the nullity of A, is determined by the rank-nullity theorem. This theorem states that the sum of the rank and the nullity of a matrix must equal the number of columns of the matrix. So, the dimension of Nul A is the number of columns (7) minus the rank (3), which gives us a nullity of 4. This tells us there are 4 dimensions in the null space of A.