Question

Consider the following statements :
I. There exists a linear transformation from \(\R^3\) to itself such that its range space and null space are the same.
II. There exists a linear transformation from \(\R^2\) to itself such that its range space and null space are the same.
Then

• A. both I and II are TRUE
• B. I is TRUE but II is FALSE
• C. II is TRUE but I is FALSE
• D. both I and II are FALSE

Answer 1

The Correct Option is C

We need to evaluate the truthfulness of each statement regarding linear transformations and their properties related to range space and null space.

Statement I: There exists a linear transformation from \(\R^3\) to itself such that its range space and null space are the same.

To understand this, we utilize the Rank-Nullity Theorem, which states:

\(\text{dim}(\text{Null}(T)) + \text{dim}(\text{Range}(T)) = \text{dim}(V) \), where \( V \) is the vector space on which the transformation \( T \) is defined.

For a linear transformation from \(\R^3\) to itself, we have:

\(\text{dim}(\text{Null}(T)) + \text{dim}(\text{Range}(T)) = 3 \)

For the null space and range space to be the same, their dimensions must be equal. Say they both have dimension \( k \), so:

\(k + k = 3 \)

which leads to \(2k = 3 \), and clearly \( k \) cannot be a fraction. Therefore, such a transformation cannot exist in \(\R^3\) .

Conclusion for I: FALSE.

Statement II: There exists a linear transformation from \(\R^2\) to itself such that its range space and null space are the same.

Applying the Rank-Nullity Theorem for \(\R^2\) :

\(\text{dim}(\text{Null}(T)) + \text{dim}(\text{Range}(T)) = 2 \)

If the dimensions are the same, say they are both \( k \), then:

\(k + k = 2 \)

giving \(k = 1 \). This is possible, where both subspaces can have the same dimension of 1. Therefore, such a transformation can indeed exist in \(\R^2\) .

Conclusion for II: TRUE.

Based on this analysis, the correct option is that "II is TRUE but I is FALSE".