All Questions
Tagged with vector-spaces proof-writing
207
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Given, a matrix representation of a linear transformation, find a formula for it and represent it by a matrix with respect to a given ordered basis.
Suppose the matrix of the linear transformation $T:\Bbb R^3\to \Bbb R^3$ with respect to the standard ordered basis of $\Bbb R^3$ is $A=\begin{pmatrix}1 && 2 && 3\\ 3&& 1 &&...
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Let $V$ be a vector space and $W_1, W_2$ be subspaces of $V.$ Show that $W_1/W_1\cap W_2\cong (W_1+W_2)/W_2.$
Let $V$ be a vector space over a field $F$ and let $W_1, W_2$ be two subspaces of $V.$ Show that $W_1/W_1\cap W_2\cong (W_1+W_2)/W_2.$ Hence conclude that, $\dim (W_1+W_2) = \dim W_1 + \dim W_2 -\dim (...
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If a vector $v_1$ from some subspace $V$ is added to another vector $v_2$ and the result is in $V$, does that imply that $v_2$ is in $V$?
I am working through a problem in Sheldon Axler's linear algebra book with subspaces. In one of my proofs I use this argument:
Given $v \in V$ and $u$ may or may not be in $V$: $$\text{if}\ v+u \in V\ ...
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Is this sufficient to show the two spaces are not isomorphic?
Yesterday I was studying about isomorphisms between vecto spaces, and basically the fundamental note I highlighted stated that "two vector spaces whose dimensions are different cannot be ...
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($L$ is an affine space of dimension one) $\Leftrightarrow$ ($L$ is a line)
Definition (I):
Let $x,y \in \mathbb{R}^n $ with $x\neq y$.
A line in $\mathbb{R}^n$ is defined as the subset $L=\{t(y-x)+x | t\in \mathbb{R}\}\subset \mathbb{R}^n$.
Statement: ($L$ is an affine ...
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Having a hard time writing a proof in Linear Algebra. [duplicate]
Let $S$ be a set of nonzero polynomials in $P(F)$ such that no two have
the same degree. Prove that $S$ is linearly independent.
I tried writing the solution as follows:
Let $g_1,g_2,...,g_n$ be some ...
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If $V$ is a subspace of a finite dimensional inner product vector space, prove that $(V ^ \perp)^\perp = V$
For any subspace $V$ of a finite dimensional inner product vector space, prove that $(V ^ \perp)^\perp = V$.
My proof is below. I request verification, critique, or improvement.
Note: Other proofs ...
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proof that the dimension of a subspace is less than or equal to the dimension of the vector space
Let $V$ be a finite-dimensional $K$-vector space and let $U$ be a subspace of $V$.
I'm trying to prove that $\dim (U) \leq \dim (V)$, but I'm not sure if the arguments what I gave was enough. Please ...
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Proof of the following statement
My proof of the above statement in the forward direction is as follows:
For linear operators we have $T_a$ is surjective iff $\text{Im}T_a=\ell^\infty$. Suppose $T_a$ is sujective which implies that $\...
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Prove that $T_a$ is a bounded linear operator on $(\ell^\infty,||\cdot||_\infty)$, and find the operator norm
I am trying to prove that $T_a$ is a bounded linear operator on $(\ell^\infty,||\cdot||_\infty)$, and that $||T_a||_{op}=||a||_\infty$ with $a = (a_n)_{n\geq 1}\in\ell^\infty$ and $T_a:\ell^\infty\to\...
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Confusion about $\{f\in C[0,1]|f(0)=0\}$ being closed in $(C[0,1],||\cdot||_\infty)$ but not closed in $(C[0,1],||\cdot||_1) $.
I am not sure as to why $Y=\{f\in C[0,1]|f(0)=0\}$ is closed with respect to the infinity norm but not the $L^1$ norm. I tried to use a counter example $f_n(t)=1-e^{-nt}$, by trying to show that the ...
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From a sequence in a Hilbert space $H$ to a countable set of linearly independent vectors.
Let $(y_n)_n$ be a sequence in $H$. Then exists an at most countable set of linearly independent vectors $(x_j)_{j\in J}$ in $H$ such that $$\text{sp}\left(y_n\;|\; n\in\mathbb{N}\right)=\text{sp}\...
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Prove that S(n,m) is an integer.
Let $\mathcal{P}_n(\mathbb{Q})$ be the polynomials of degree at most $n$ with rational coefficients. Let $\gamma=\left((x)_0, \ldots,(x)_n\right)$ be the list of polynomials defined inductively by $(x)...
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Trying to prove this relationship. Not sure if this is correct.
I am trying to prove whether or not that:
The triangle inequality holds; that is, for every vectors $x$ and $y$, $$||x+y|| \leq ||x|| + ||y||,$$ for the vector function $$||x|| = \sum_{i=1}^n x^2_i.$...
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$(F^\infty, F,+,\cdot ,\times)$ is commutative linear algebra with identity over $F$
Definition: Let $(V,F,+,\cdot)$ be a vector space over $F$. $(V,F,+,\cdot,\times)$ is linear algebra over $F$ if $\times:V\times V\to V$ have following properties:
$(1)$ $\alpha \times (\beta \times \...
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