[Community Question] Linear-algebra: T/F: The Range of a Linear Transformation must be a subset of the domain.

One of our user asked:

I am having difficulty understanding why it is false. As I understand, the range is the output of everything from the domain into the transformation function, or the result of the domain being outputted to the co-domain via the transformation. Essentially, I cannot think of a counterexample where the range of a linear transformation is not in the domain. I would appreciate if anyone could shed some light on this basic concept that I am struggling with.

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Calculus: Intersection of hyperplanes in $\mathbb{R}^5$

Consider the $4$ hyperplanes in $\mathbb{R}^5$ given by the equations $$x_1-x_2+x_3+x_4+2x_5=0$$ $$2x_1-x_2+6x_3+2x_4+6x_5=0$$ $$3x_1-3x_2+3x_3+4x_4+7x_5=0$$ $$x_1+x_2+9x_3+x_4+6x_5=0$$ Let $V \leq \mathbb{R}^5$ be the intersection of these hyperplanes. Find a basis for $V$ .

[Community Question] Calculus: calculating 2 constants in a function

One of our user asked: $$M=\{f\in C[0,2\pi],\int_{0}^{2\pi}f(x)sinxdx=\pi,\int_{0}^{2\pi}f(x)sin2xdx=2\pi\} $$ $a,b\in \mathbb R, g\in M, g(x)=asinx+bsin2x,x\in [0,2\pi]$ I've read on the answers that $a=1,b=2$ and I don't know how to calculate them. Can somebody explain me,please? By the way, the problem is to determine $ \int_{0}^{2\pi}(g(x))^2 dx$ so you have to first get the constants $a$ and $b$ .

Calculus: A difficulty in understanding a step in a solution.

Here is the solution: But I could not understand how the last term in the fourth line came from the line before it, could anyone explain this for me please?

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