Math Snap

STEP 1

1. The matrix $A$ is a $7 \times 5$ matrix.
2. The null space of $A$ is 4-dimensional.
3. The rank-nullity theorem applies, which states that for a matrix $A$ with $n$ columns, $\text{rank}(A) + \text{nullity}(A) = n$.

STEP 2

1. Use the rank-nullity theorem to find the rank of $A$.
2. Determine $\operatorname{dim} \operatorname{Row} A$.
3. Determine $\operatorname{dim} \operatorname{Col} A$.
4. Compare results with the given options.

STEP 3

Apply the rank-nullity theorem. For a $7 \times 5$ matrix $A$, the number of columns $n = 5$. The nullity of $A$ is given as 4. Therefore, we can write:
$$\text{rank}(A) + \text{nullity}(A) = n$$ $$\text{rank}(A) + 4 = 5$$ Solve for $\text{rank}(A)$:
$$\text{rank}(A) = 5 - 4 = 1$$

STEP 4

The dimension of the row space of $A$ is equal to the rank of $A$. Therefore:
$$\operatorname{dim} \operatorname{Row} A = \text{rank}(A) = 1$$

STEP 5

The dimension of the column space of $A$ is also equal to the rank of $A$. Therefore:
$$\operatorname{dim} \operatorname{Col} A = \text{rank}(A) = 1$$

Compare the results with the given options:
- rank $A = 1$
- $\operatorname{dim} \operatorname{Row} A = 1$
- $\operatorname{dim} \operatorname{Col} A = 1$
The correct option is:
D. $\operatorname{rank} A=1, \operatorname{dim} \operatorname{Row} A=1, \operatorname{dim} \operatorname{Col} A=1$
The correct answer is:
$$\boxed{D}$$