Skip to main content

Calculus: perpendicular planes in $\mathbb{R}^4$

Let $W \leq \mathbb{R}^n$ and set $$W^{\perp} = \{v \in \mathbb{R}^n | v.w =0, \hspace{0.25cm} \forall w \in W\}$$ (a) Show that $W^{\perp} \leq \mathbb{R}^n$.

(b) Let $v_1=\langle 1, 1, 1, 2\rangle$,$v_2=\langle 2, 1, 1, 1\rangle$, and $v_3=\langle -1, 2, 2, 7\rangle$. Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^4$ be the transformation given by $e_1 \mapsto v_1$, $e_2 \mapsto v_2$, and $e_3 \mapsto v_3$.Let $W=img(T)$. Find a basis for $W$ and $W^{\perp}$.

Labels:

Comments

Post a Comment

Popular posts from this blog

[Community Question] Linear-algebra: Are linear transformations between infinite dimensional vector spaces always differentiable?

One of our user asked: In class we saw that every linear transformation is differentiable (since there's always a linear approximation for them) and we also saw that a differentiable function must be continuous, so it must be true that all linear operators are continuous, however, I just read that between infinite dimensional vector spaces this is not necessarily true. I would like to know where's the flaw in my reasoning (I suspect that linear transformations between infinite dimensional vector spaces are not always differentiable).
Post a Comment
Read more

[Community Question] Calculus: prove $\int_0^\infty \frac{\log^2(x)}{x^2+1}\mathrm dx=\frac{\pi^3}{8}$ with real methods

One of our user asked: I am attempting to prove that $$J=\int_0^\infty\frac{\log^2(x)}{x^2+1}\mathrm dx=\frac{\pi^3}8$$ With real methods because I do not know complex analysis. I have started with the substitution $x=\tan u$ : $$J=\int_0^{\pi/2}\log^2(\tan x)\mathrm dx$$ $$J=\int_0^{\pi/2}\log^2(\cos x)\mathrm dx-2\int_{0}^{\pi/2}\log(\cos x)\log(\sin x)\mathrm dx+\int_0^{\pi/2}\log^2(\sin x)\mathrm dx$$ But frankly, this is basically worse. Could I have some help? Thanks.
Post a Comment
Read more