We are given the following problem:     Let $S$ be the set of all functions $y$ that satisfy the following differential equation   $$2\dfrac{d^2y}{dx^2}  - 3\dfrac{dy}{dx} + y = 0.$$   Show that $S$ is a subspace of the vector space $A$, where $A$ is the set of all functions $f : \mathbb{R} \rightarrow \mathbb{R}$.   I do not know how to approach the problem.
120924
$V$ is space of differentiable functions $f(0,1) \to \mathbb{R}$ and $W$ is a subset of $f$ that meets $f'(x) - 3f(x) = 0$ for all $x\in (0,1).$  Is subset $W$ a subspace of $V$?  I know that I have to prove that it's closed under scalar multiplication and under addition. But I forgot all the stuff about differentiable functions, so I don't know how to do this. But my intuition tells me that if there's a zero it's valid subspace :) If anyone could elaborate on this I would be grateful.
65114
Let $ Y = \{ (1,1),(0,0\} \subset \mathbb A_k^2$. Find the $ \mathbb I(Y)$.   As if $f(x,y) \in \mathbb I(Y)$ then as $f(0,0)=0$ hence constant term of $f$ is zero and as $f(1,1)=0$ therefore sum of the coefficients of $f(x,y)$ is zero. But how can we describe this set properly?
108687
Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite number of them? I have seen sometimes in pure maths that you take an arbitrary finite collection of sets and show the statement you are trying to prove is true for them to deduce it is true for the whole infinite collection. Why is this technique allowed? Can I do it for this case? Why?
101776
Determine whether the set $W=\{(2a-2,3b,2a-3b)\}$ is a subspace in $\mathbb{R}^{3}$. Describe the set.  I have tried putting the set into matrix form but don't know which is correct. I also know that it needs to be closed under addition and multiplication but can't figure out how to show that.
13903
I want to see if my proof is true or I thought very trivially?    If $H$ is a subspace of a finite dimensional vector space $V$, show there is a subspace $K$ such that $H\cap K=0$ and $H+K=V$    So far I have tried :    $H\subseteq V$ is a subspace $\Rightarrow\,\exists\;K=(V-H)\subseteq V$ $K$ is a subspace because it's the sum of two subspace $V$ and $(-H)$ Now we should prove that $H\cap K=0$ We assume $H\cap K\neq0\Rightarrow\exists\,\vec u\neq0\,,\,\vec u\in H\cap K$ $\vec u\in H\cap K\Rightarrow \begin{cases}\vec u\in H\\\vec u\in K\,\Rightarrow\vec u\in V-H\Rightarrow\vec u\in V,\,\vec u\notin H\end{cases}$ That's a contradiction so our hypothesis was wrong and $\vec u=0$
35244
I want to find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in\mathbb{R}$: $$f(x^2+y+f(y))=2y+f(x)^2.$$  Suppose we have given a solution $f$. Then $f$ is surjective (set $x=0$). Moreover, I have shown that $f(-x)^2=f(x)^2$ for all $x\in\mathbb{R}$ as well as $f(x)=0\Rightarrow x=0$. But I could not make any progress.  I tried for example $y:=-f(x)^2/2$. Then we get $x^2-f(x)^2/2+f(-f(x)^2/2))=0$. Can someone tell me how to proceed?  Best wishes
19147
I am self-studying  Hoffman and Kunze's book Linear Algebra. This is exercise $1$ from page $111$.  If $W$ is a subset of a vector space $V$, we define $W^{0}=\{f\in V^{\star}|f(w)=0 \operatorname{for all}w\in W\}.$     Let $n$ be a positive integer and $\mathbb{F}$ be a field. Let $W$ be   a set of all vectors $(x_{1},\ldots,x_{n})$ in $\mathbb{F}^{n}$ such   that $x_{1}+\cdots+x_{n}=0$.      a) Prove that $W^{0}$ consists of all linear functionals $f$ of the   form $$f(x_{1},\ldots,x_{n})=c(\sum_{j=1}^{n}x_{j}).$$      b) Show that the dual space $W^{\star}$ of $W$ can be "naturally"   identified with the linear functionals   $$f(x_{1},\ldots,x_{n})=c_{1}x_{1}+\cdots+c_{n}x_{n}$$ on   $\mathbb{F}^{n}$ which satisfy $c_{1}+\cdots+c_{n}=0$.   My approach for part a) First, we note that $e_{i}-e_{j}\in W$ for all $i,j\in\{1,2,\ldots,n\}$. If $f\in W^{0},$ then we have $f(e_{i})-f(e_{j})=f(e_{i}-e_{j})=0.$ Therefore $f(e_{i})=f(e_{j})$ and we have $f(e_{i})=c$, for all $i\in \{1,2,\ldots,n\}$, where $c$ is a constant. Therefore, if $w=(x_{1},\ldots,x_{n})\in W$, then $f(x_{1},\ldots,x_{n})=c(\sum_{j=1}^{n}x_{j}).$ The converse is easy.   I was not able to solve the part b).
36881
Show that $P_n = \{$Polynomials with real coefficients of degree $≤ n\}$ is an $(n+1)$-dimensional subspace of the infinite-dimensional vector space of all real polynomials.
7683
Let $V$ and $W$ be vector spaces over a field $F$. Let $L(V,W) = \{T:V\to W : T \text{ is linear} \}$, that is, $L(V,W)$ is the collection of all linear functions from $V$ to $W$. For $S,T \in L(V,W)$ and $a \in F$, define new functions $(S+T):V \to W$ and $ aT :V \to W$ by $(S+T)(x) = S(x)+T(x)$ ; $(aT)(x) = a(T(x))$.   Prove that $S+T$ belongs to $L(V,W)$ and $aT$ belongs to $L(V,W)$.
98534
I don't know how to interpret this function:     Let $D$ be an open subset of $\mathbb{R}^2$ with a continuous function $f:D \rightarrow \mathbb{R}$  and   \begin{align} y'(x)=f(x,y(x)) \end{align}   is a continuous, explicit first-order differential equation defined on $D$.   From $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ I interpret $f$ as a two variable function, $f(x,y)$.  But here $y$ is a function of $x$, so isn't $f(x,y(x))$ a function of one variable, $f:\mathbb{R} \rightarrow \mathbb{R}$?
62659
I have a conjecture. I have a problem proving or disproving it.      Let $w \in \Lambda^k(V)$ be a $k$-vector. Then $W_w=\{v\in V: v\wedge w = 0 \}$ is a $k$-dimensional vector space if and only if $w$ is decomposable.   For example, for $u=e_1\wedge e_2 + e_3 \wedge e_4$ we have $W_u = 0$.
63325
May I please ask how to show the dual space of the vector space of all polynomials is isomorphic to the infinite-dimensional Euclidean vector space over the reals?  (i.e. Show that $(\Bbb{R}[X])^*$ and $\Bbb{R}^{∞}$ are isomorphic where $\Bbb{R}$ means the set of all reals)
83600
For $n \in \mathbb{N}$ and $W \leq \mathbb{F}^n$, show that there exists a homogeneous system of linear equations whose solution space is $W$.  Here's my work: Since $W \leq \mathbb{F}^n$, $k = dim(W) \leq dim(\mathbb{F}^n)$. Let's say that $\{w_1,w_2,...,w_k\}$ is a basis of $W$. Now, construct a matrix $A$ (of size $k \times n$) such that its rows are elements from the basis of $W$, stacked together. The row space of $A$ is $W$, so the row space of its row-echelon form is $W$ too. At this point, I'm stuck! I'm trying to come up with a homogeneous system with the help of $A$, though there may exist other easier ways of approaching this problem. Could someone show me the light? P.S. $W \leq \mathbb{F}^n$ stands for $W$ is a subspace of $\mathbb{F}^n$. P.P.S. Isn't this equivalent to saying that $W$ is the null-space of some matrix? Can we go ahead along these lines, and construct a matrix $P$ such that $Pw = 0$ for all $w \in W$?
58021
we consider the problem $$ \begin{cases} x^2 y''+xy'+\lambda y\\ y(1)=y(e)=0 \end{cases} $$ The question is to find the eigenvalues and eigenfunctions for this problem.  I try to put $y(x)=x^r$ and plug it in the equation, then we obtain the caracteristic equation $$r^2+r+\lambda =0$$ In the case $\lambda &gt;0$ we put $\lambda= \alpha^2$ where $\alpha \in \mathbb{R}^\star_+$. Then the caracteristic equation is  $$ r^2+\lambda=0 $$  Case 1. If $\lambda &gt;0$ We put $\lambda = \alpha^2$ where $\alpha \in \mathbb{R}^\star_+$.  Then the genral solution of the equation is  $$ y(x)=C_1 \cos(\alpha x)+ C_2 \sin(\alpha x). $$ We have $y(1)=0 =&gt; C_1=0$ and $y(e)=0 =&gt; C_2 \sin(\alpha e)=0$  $Sin(\alpha e)=0 =&gt; \alpha e = \pi + 2 k \pi, \ k \in \mathbb{Z}^\star$  My solution is correct?
84836
Could any one just tell me a sketch of the proof of the followings?  $1$. If $n&gt;1$, does every linear transformation on $\mathbb{R}^n$ have an invariant subspaces of dimension $2$?  $2$ Is there a vector space $V$ and a linear transformation $T$ on $V$ such that it has exactly $3$ invariant subspace?  Thank you!
103477
Let $F(\mathbb{R},\mathbb{R})$ be the vector space of all functions from $\mathbb{R}$ to $\mathbb{R}$. Prove that for all $n\in\mathbb{N}$, the set of vectors $$\left\{sin(x), sin(2x),sin(2^2x),\dots,sin(2^nx)\right\}$$   form a linearly independent subset of $F(\mathbb{R},\mathbb{R})$.  I know that to form a linearly independent subset the set must be closed under addition, closed under scalar multiplication, and contain the neutral element.  Showing that it contains the neutral element is relativley straight forward since when $x=0$ every element also equals $0$. Therefore this subset contains the neutral element. However, I am having a hard time coming up with a way to approach the parts showing that it is closed under scalar multiplication and addition for all $x$. Do you have a piece of advice or a hint on how to approach those part?
13364
I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated.