Final answer:
The subset W of upper triangular 3×3 matrices is a subspace of V, the vector space of all 3×3 matrices, since it contains the zero vector and is closed under vector addition and scalar multiplication.
Step-by-step explanation:
The given subset W, which is composed of all upper triangular 3×3 matrices, is a candidate to be a subspace of the larger vector space V which consists of all 3×3 matrices.
To verify whether W is a subspace of V, we need to confirm that it satisfies three main properties: it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication.
Let's consider the zero vector in this context, which is the 3×3 matrix where all entries are zero. This matrix is trivially upper triangular, so W contains the zero vector.
Next, the addition of vectors (matrices in this context) is commutative as mentioned, meaning that if A and B are upper triangular matrices, their sum A + B is also upper triangular, so W is closed under addition.
Lastly, scalar multiplication of an upper triangular matrix by any real number will result in another upper triangular matrix, which demonstrates that W is closed under scalar multiplication.
Since all three conditions are satisfied, we can conclude that W is indeed a subspace of V, the vector space of all 3×3 matrices.
The complete question is:Is the subset W, which is equal to an upper triangular 3×3 matrices, ⎣#00⎦ ⎣##0⎦ ⎣###⎦, a subspace of vector space V, which is equal to all 3×3 matrices? Support your answer to show why or why not? is: