# True/False Linear Algebra Vector Spaces/Subspaces

I was doing a bunch of true false for this section and here are a couple I can't seem to understand.

• The solutions of a matrix equations $Ax=0$ forms a vector space.
• The set of nonsingular $n\times n$ matrices forms a vector space.
• The set of degree four polynomials forms a vector space
• The set of vectors of length 1 forms a subspace of $\mathbb R^2$.
• The set of all functions $f(x)$ such that $f(0)=0$ forms a vector space.

Could you please leave an explanation if you can? Thanks

• Most of these statements are incredibly imprecise. For example, statements 2 and 3 are false if read literally, because vector spaces carry more structure than sets. So "The set [of whatever] is a vector space" is always false. What the sentence means is: "The set [of whatever] with vector addition and scalar multiplication defined in such-and-such a way is a vector space." Or: "The set [of whatever] can be made into a vector space." I mention this because you can make the same set into a vector space in different ways. – symplectomorphic Jan 27 '14 at 0:04
• this is what I have so far are these correct? 1. False 2. True 3. False 4. False 5. False – user123204 Jan 27 '14 at 0:30
• No. Some of these are wrong. But if you want help, you have to explain your thinking. The people on this forum won't do your HW for you. – symplectomorphic Jan 27 '14 at 0:39
• ok I think the first one is true because if x1 and x2 are 0. the addition and scalar properties are true. Also any scalar would work. I think the second one is false because the zero matrix has a 0 determinant so its singular and is a vector space. I think the 3rd is true because any polynomial can be added together and multiplied by a scalar regardless of degree. I think the 4th one is false because the zero vector has length 0 but it forms a subspace of r2. the vector <6,8> does not necessarily form a subspace. Finally I am prety sure the last one is false – user123204 Jan 27 '14 at 0:54
• because you can't guarantee that addition axiom is true. – user123204 Jan 27 '14 at 0:55

Can you show that if $Ax=0$ and $Ay=0$, then $A(x+y)=0$ and $A(\lambda x)=0$ for any $\lambda$?

Is the zero matrix invertible?

Is the zero polynomial of degree four?

Does the zero vector have length $1$?

Can you show that if $f,g$ are $0$ at zero then their sum $f+g$ is $0$ at zero and for any $\lambda$, $\lambda f$ is zero at $0$?

• So the second one would be false because the determinant is 0? – user123204 Jan 26 '14 at 23:58
• @user123204 I think you can answer that yourself. – Pedro Tamaroff Jan 27 '14 at 0:01
• I am still confused... I know the zero matrix is not invertible, zero vector has length 0 and just because f(0)=0 that doesn't mean f+g=0 and the zero polynomial is degree undefined – user123204 Jan 27 '14 at 0:08
• Pedro's hints are trying to remind you that in order for a set to be a vector space, it needs some notion of a zero vector. – symplectomorphic Jan 27 '14 at 0:26