# Tagged Questions

18 views

### Understanding the operator of differentiation on the vector space of polynomials

I have been reading through Linear Algebra Done Right by Sheldon Axler. The book defines an operator as a linear map from a vector space to itself. It then considers at another part of the book the ...
40 views

### What is wrong with this proof that if $V = U_1 \oplus W$ and if $V = U_2 \oplus W$, then $U_1 = U_2$?

Claim: Let $U_1, U_2$ and $W$ be subspaces of a vector space $V$. Suppose $V = U_1 \oplus W$ and $V = U_2 \oplus W$. Then $U_1 = U_2$. "Proof" Let $v \in V$. Then $\exists \space u_1 \in U_1$ ...
38 views

### Problems on vector spaces

Let $E$ a $\mathbb{K}$-vector space of finite dimension $n$, $\mathcal{V}$ a subspace of $\mathcal{L}(E)$ such that $$\forall u\in\mathcal{V}\setminus \{0\},u\in\mathcal{GL}(E)$$ a) Show that ...
22 views

### Solution to homogeneous linear differential equation form a vector space

Show that the solutions of a homogeneous linear differential equation $y"+a(x)y'+b(x)y = 0$ form a vector space. What is its dimension? I understand that the dimension is 2 and that 0 is a solution ...
16 views

### Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
29 views

### Subsets that are also vector spaces

The vector space $R^3$ and the subset M consists of the vectors $(\xi_1,\xi_2,\xi_3)$ for which i) $\xi_1 = 0$ ii) $\xi_1 = 0$ or $\xi_2 = 0$ iii) $\xi_1 + \xi_2 = 0$ iv) $\xi_1 + \xi_2 = 1$ ...
22 views

### Function from one Null space to Another

Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is ...