Let V be the vector space of 4 X 4 real matrices and let U be the subspace of all matrices B in V such that 
trace(AB) = 0 for all diagonal matrices A. Find the dimension of U.

We are given that \( V \) is the vector space of all \( 4 \times 4 \) real matrices, and \( U \) is a subspace of \( V \) defined by the condition that:

\[
\text{trace}(AB) = 0 \quad \text{for all diagonal matrices} \ A.
\]

### Step 1: Express the Trace Condition
Let \( B \) be any \( 4 \times 4 \) matrix, and \( A \) be any diagonal \( 4 \times 4 \) matrix. Denote the diagonal entries of \( A \) by \( a_1, a_2, a_3, a_4 \). The trace of the matrix product \( AB \) is:

\[
\text{trace}(AB) = a_1 B_{11} + a_2 B_{22} + a_3 B_{33} + a_4 B_{44}.
\]

For \( B \) to belong to \( U \), we require that \( \text{trace}(AB) = 0 \) for all choices of \( a_1, a_2, a_3, a_4 \). This means that:

\[
a_1 B_{11} + a_2 B_{22} + a_3 B_{33} + a_4 B_{44} = 0.
\]

Since this equation must hold for all values of \( a_1, a_2, a_3, a_4 \), we conclude that:

\[
B_{11} = B_{22} = B_{33} = B_{44} = 0.
\]

Thus, all the diagonal entries of \( B \) must be zero. Therefore, the subspace \( U \) consists of all matrices \( B \) with zero diagonal entries.

### Step 2: Dimension of Subspace \( U \)
A general \( 4 \times 4 \) matrix has 16 entries, but the condition that all diagonal entries are zero removes 4 degrees of freedom (corresponding to the 4 diagonal entries). Thus, the matrices in \( U \) have non-zero entries only in the off-diagonal positions.

The number of off-diagonal entries in a \( 4 \times 4 \) matrix is \( 16 - 4 = 12 \). Therefore, the dimension of the subspace \( U \) is 12.

### Conclusion
The dimension of the subspace \( U \) is:

\[
\boxed{12}.
\]

Do you want further details or have any questions?

Here are 5 related questions:
1. How do you determine the dimension of a vector space of \( n \times n \) matrices?
2. What is the trace of a matrix, and why is it important in linear algebra?
3. Can you describe the structure of diagonal matrices and their properties?
4. How does the condition \( \text{trace}(AB) = 0 \) restrict the possible matrices \( B \)?
5. What is a subspace, and what properties must it satisfy?

**Tip:** The trace function is a linear map, and its kernel can be useful when identifying subspaces related to trace conditions.

Let V be the vector space of 4 X 4 real matrices and let U be the subspace of all matrices B in V such that trace(AB) = 0 for all diagonal matrices A. Find the dimension of U.

Explore how to find the dimension of subspace U in the vector space of 4x4 real matrices, where the trace of AB is zero for all diagonal matrices A. This problem introduces important linear algebra concepts, including vector spaces, subspaces, and the trace of a matrix. Suitable for undergraduate students.

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