ntabb1

2022-03-31

Determine whether the given set S is a subspace of the Vector space V.

(there are multiple)

alenahelenash

Expert2023-04-26Added 556 answers

To determine whether the set S of all upper triangular matrices is a subspace of the vector space V of all n-by-n matrices with real entries, we need to check if S satisfies the three conditions of being a subspace:

1. The zero vector of V is in S.

2. S is closed under vector addition.

3. S is closed under scalar multiplication.

Let's consider each of these conditions in turn:

1. The zero vector of V is the matrix of all zeros, which is clearly an upper triangular matrix. Therefore, the zero vector is in S.

2. To show that S is closed under vector addition, we need to show that if A and B are upper triangular matrices, then A + B is also an upper triangular matrix. This is true because the sum of two upper triangular matrices is also an upper triangular matrix. Therefore, S is closed under vector addition.

3. To show that S is closed under scalar multiplication, we need to show that if A is an upper triangular matrix and k is a scalar, then kA is also an upper triangular matrix. This is true because multiplying an upper triangular matrix by a scalar does not change its upper triangular structure. Therefore, S is closed under scalar multiplication.

Since S satisfies all three conditions of being a subspace, we can conclude that S is a subspace of the vector space V.

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