# Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

82,523 questions
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### Change of basis of a lattice

Let $\Gamma:=\Big\{(i, 0), (0,i), (\sqrt 3,\sqrt 3)\Big\}$ be a basis of a lattice in $\mathbb C^2$. Can we find another basis $\Gamma':=\{(1,0),(0,1), w\}$ such that span over $\mathbb R$ of $\Gamma$...
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### The number of scalar additions required to compute P(QR)

Let P, Q, R be matrices of order $3\times5, 5\times7$ and $7\times3$ respectively. What is the number of scalar additions requiered to compute P(QR)? Do we have any formula to compute this? If we had ...
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### Get rectangular coordinates of a 3d point, with the polar coordinates?

Ok, so imagine a cannon on an cartesian space. X and Y are sideways and Z is upwards. It can rotate horizontally on the world's XY axis (right is clockwise, and left is counter-clockwise), and also ...
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### Endomorphism $\phi: M_{n}(\mathbb{C}) \rightarrow M_{n}(\mathbb{C})$ that stabilizes $GL_{n}(\mathbb{C}).$

I've been given this as a homework assignment, and have no idea how to proceed. Can anyone help? The question is: (1) Let $\phi: M_{n}(\mathbb{C}) \rightarrow M_{n}(\mathbb{C})$ be an endomorphism ...
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### Finding a solution for $x^2+y^2-z^2 = 1$

Let $$K = \{ (x,y,z) \in \mathbb R^3 \mid x^2 + y^2 - z^2 = 1 \}$$ Prove that $\mbox{Span}(K) = \mathbb R^3$. How can I solve this and find the vectors that will fit it?
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### homomorphism question

Check $p(x) + p(-x) ∈ P^e$ for every $p(x) ∈ \mathbb{R}[x]$. Check that the map $Ψ:\mathbb{R}[x]\to P^e$ given by $Ψ(p(x))=(p(x)+p(-x))/2$ is a linear map. Further check that $Ψ^2=Ψ$. Determine ...
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### Proving $A^n=\left[\begin{smallmatrix}1&(2^n-1)a\\0&2^n\end{smallmatrix}\right]$ [on hold]

I have this input \begin{bmatrix}1&a\\0&2\end{bmatrix} and I have no clue on how to prove it right. All I could do was to try with $A^2$, $A^3$ and so on but I can't prove it.
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### Simultaneous equations result in decimal answer?

I have these 2 equations. $$\left\{ \begin{split} y &= 2x -2 \\ 3y &= -2x + 6 \end{split}\right.$$ I have worked it all out, and plotted the graphs and the point they meet is: $(x,y)=(1,1.5)$....
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### fill in table - non-isomorphic groups - permutation.

I am a student in computer science - first year. I study linear linear algebra 2 - course of linear algebra 1. - In some institutions academic studies teach the courses together / teach in another way....
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### Is it possible to determine the constituents of a matrix product, given the result?

Suppose we have a set $M$ of two or more matrices such that every matrix product $X$ composed of matrices drawn with replacement from $M$ is unique. Is there a set $M$ for which we can determine the ...
Consider $h : \mathbb{R}^+ \to \mathbb{R}$ given as $h(\alpha) = U \{A^2 + 2\alpha AF + \alpha^2 F F^T\}^{-1} U^T$ $U$ is $1\times m$ $A$ is $m\times m$ symmetric PSD. $F$ is $m\times m$ diagonal ...
STATEMENT: Consider two linear maps $q_1,q_2:V \rightarrow V$ such that $q_1\circ q_1=q_1$ and $q_2\circ q_2=q_2$. Assume that $q_1\circ q_2=q_2\circ q_1$. Show that $q_2($ker $q_1)\subseteq$ ker $q_1$...