# 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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### How do I determine whether a set spans or does not span a vector space?

$\{1+x^2,1+x+2x^2,x+x^2\}$ in $P_2$ Does the set span $P_2$? I understand that the set is linearly dependent but how can I prove that the two independent elements cannot form every element in $P_2$? ...
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### Linear Map is Homeomorphic in $\mathbb R^k$

I am having trouble understanding a proof in Rudin's "Real and Complex Analysis." The theorem states that To every linear transformation $T$ of $\mathbb R^k$ into $\mathbb R^k$ corresponds a real ...
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### Show that $A$ and $A^T$ do not have the same eigenvectors in general

I understood that $A$ and $A^T$ have the same eigenvalues, since $$\det(A - \lambda I)= \det(A^T - \lambda I) = \det(A - \lambda I)^T$$ The problem is to show that $A$ and $A^T$ do not have the same ...
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### Splitting condition for iteration method

In iteration methods to solve $Ax = b$ we have the standard form as $$M x^{(k+1)} = N x^{(k)} + b \tag{*}$$ which $A = M - N$, and $M^{-1}N$ is called iteration matrix. The standard convergence ...
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### find example of linear transformation

Q: let $n \geqslant 3$, find an example of linear transformation $T: R^n \to R^n$ that $Im \cap ker T =\{0\}$ and that $\text{ker}\ T=\{(x_1,x_2,...,x_n) | x_1+x_2+...+x_{n-2}=0 \}$. I'm pretty sure ...
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### If $\vec{v}$ is an eigenvector of $A$, then also $B\vec{v}$, when $AB = BA$ [duplicate]

I have the following problem: Let $V$ be a finite dimensional vector space. Let $A$, $B$ be linear maps of $V$ into itself. Assume that $AB = BA$. Show that if $\vec{v}$ is an eigenvector of $A$, ...
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### Kernel of a matrix pencil

Let $A,B$ be $n\times n$ singular real matrices such $ker A\cap ker B=\{0\}$, how could I show that there exists $x\in \mathbb R$ such that $ker (A+xB)=\{0\}$?
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### Determining span, is there an easier way to remember it?

As I understand it, to set up a problem to determine if the vector spans $\mathbb{R}^n$ or if the given vector is in the Span of $$(v_1,v_2,...,v_n)$$ you take the vector and set up an augmented ...
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### quadratic equation plot investigation

Let $f(x) =-x^2-4x+18$ so i plot it like this : But my imagination created the following: $-x^2-4x+18=0 -> x^2+4x = 18-> x^2+4x-18 = 0$ Which yields the parabola upside down. Where's the ...
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### Minimum distance between Vector and its projection

I'm struggling to get to an easy and simple algebraic solution for this question: I actually thought about optimization with the Cauchy-Schwarz Inequality but it just got dirty and probably wrong. ...
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### Orientations of $n$-dimensional vector spaces

Let $V$ be a $n$-dimensional vector space. An $n$-tuple $(\mathbf{a_1},...,\mathbf{a_n})$ of independent vectors in $V$ is called an $n$-frame in $V$. In $\mathbb{R}^n$ we call such a frame right-...