Let A be a 3 x 5 matrix. If dim(null(A))=2, then the dimension of the column space of A isSelect one:a. 3b. 1c. 2d. None of the other choices is correct
Question
Let A be a 3 x 5 matrix.
If dim(null(A))=2, then the dimension of the column space of A is
Select one:
- a. 3
- b. 1
- c. 2
- d. None of the other choices is correct
Solution
The dimension of the column space of a matrix is also known as the rank of the matrix. The Rank-Nullity Theorem states that the rank of a matrix plus the dimension of its null space is equal to the number of columns in the matrix.
In this case, we are given that the matrix A is a 3 x 5 matrix, meaning it has 5 columns. We are also given that the dimension of the null space of A is 2.
Using the Rank-Nullity Theorem, we can find the rank of the matrix (which is the dimension of the column space) by subtracting the dimension of the null space from the number of columns.
So, the rank of the matrix A (or the dimension of the column space of A) is 5 - 2 = 3.
Therefore, the correct answer is a. 3.
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