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
Best Answer
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 $0$?