Discussion
The dimension of the vector space over R of all 5 × 5 upper triangular matrices with complex entries is :
Answer: A
The dimension is 30.
A 5 × 5 upper triangular matrix has its entries on and above the diagonal free and the rest zero. The number of such positions is
\[ 5 + 4 + 3 + 2 + 1 = 15 \]
Each position holds a complex number, and over R each complex entry \(a + ib\) needs two real coordinates. So the real dimension is \(2 \times 15 = 30\).
15 is the dimension over C, the usual slip. 25 counts all entries of a 5 × 5 matrix, and 20 corresponds to no natural count here.
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