# Tagged Questions

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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### Confused about matrix and vector operations, properties.

I am currently studying topics in Machine Learning and came across a solution I do not fully understand. The problem #4a, the statement and solution can be found here: http://cs229.stanford.edu/...
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### Bisector of two lines in the euclidean space $\mathbb{E}_3$

Let $$r: \begin{cases} x + z = 0 \\ y + z + 1 = 0\end{cases}$$ and $$s: \begin{cases} x - y - 1 = 0 \\ 2x - z -1 = 0\end{cases}$$ be two lines in the euclidean space $\mathbb{E}_3$. It is easily ...
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### Why can the function $f(x)=||A\vec{x}-\vec{b}||^2$ be rewritten as $\vec{x}^tA^tA\vec{x}−\vec{x}^tA^t\vec{b}−\vec{b}^tA\vec{x}+||\vec{b}||^2$

Someone answered a question introducing this transformation of the function $f(x)=||A\vec{x}-\vec{b}||^2$ ; but I cannot get the idea why and how. Looks a bit like a binomial expansion, but I can't ...
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### Is a subset of an inner product space also an inner product space?

My question may seem trivial but it's important that I know this. I know for a fact that a subspace of an inner product space is also an inner product space, but how about an arbitrary subset? Could I ...
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### Prove that matrix $[S]$ associated to operator is such that $A |\zeta|^2\leq s_{ij}(x) \zeta_i \zeta_j\leq B |\zeta|^2$.

Let us consider $N\times N$ matrix $[S]$ associated to operator $S:V\rightarrow V$ where $V$ is a Hilbert space; $S$ is linear, bounded, invertible, positive and self-adjoint. Prove that $[S]$ is ...
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### Why is $V^{\perp}={0}$

Let $V$ be an inner product space. I have read a statement saying $V^{\perp}=\{0\}$. Why is this true? It seems trivial to even define an orthogonal complement to $V^{\perp}$ if it is always just $0$. ...
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### Proving two planes are parallel (question about the equation)

If I have two planes: $$5x + y - z = 7$$ $$-25x -5y + 5z = 9$$ I can see that from the first plane I get the vector $\langle5,1,-1\rangle$ from the coefficients and then from the second plane I get ...
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### Integers $a$ for which the equation $\big\lvert 6\lvert x\rvert -8\big\rvert = a+x$ has the most solutions

$$\big\lvert 6\lvert x\rvert -8\big\rvert = a+x$$ I know this should be done graphically, looking at each case and seeing for which $a$ will it intersect the $x$-axis the most times, but I can't seem ...
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### Positive linear combinations of intervals

Given two intervals at $i\in\{0,1\}$ $I_i=[-a_i,a_i]$ where $0<a_0<a_1=1-a_0<1$ and a third interval $I=[-a,a]$ where $0<a<\frac{1}2$, when is there an $\alpha,\beta\in\Bbb R$ such that ...
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