Why does the 2x2 matrix with a trace equal to 1 not contain any zero vectors?

Question

The trace of a square n×n matrix A=(aij) is the sum a11+a22+⋯+ann of the entries on its main diagonal.

Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 matrices with real entries that have trace 1. Is H a subspace of the vector space V?

Does H contain the zero vector of V?

H does not contain the zero vector of V

Hello, I was reviewing my homework problems and I can't seem to understand the logic behind these correct answers.

The matrix <[1, 0], [0, 0]> has a trace of 1, is 2x2, and uses real entries while having a zero vector. Why is the answer 'does not contain a zero vector'?

Does the zero vector mean zero's in the entire matrix?

I was under the impression that a matrix can be broken up into n vectors so [1,0] is one vector and [0,0] is another vector thereby meaning that there is a zero vector.

Answer 1

The zero vector of your vector space is the $2$ by $2$ matrix whose entries are all zeros. The trace of such matrix is zero not one.

Thus $H$ is not a subspace.

Note that $H$ is not closed under addition or scalar multiplication because the trace is not preserved under these oprations.

Answer 2

Notice the difference between the

"geometrical notion" of vector, which is a tuple with numbers in it, for example $(2,3) $ and the

"algebra notion" of vector, which is any element $v $ in a vector space $V $.

In this case, you vector space is filled with matrices, so they are your vectors. Of course one can think of a matrix as being composed of vectors, in the "geometrical notion".

In the algebra side of things, the "zero vector" is the element which does nothing when added to other elements, which is of course the matrix that only has 0s.