# 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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### Addition of Kronecker Product Matrices

Summary: Is it possible to write $A_1 \otimes A_2 + B_1 \otimes B_2$ as some object which has nice properties again, preferably as a Kronecker product itself? Each of the matrices $A_i$, $B_i$ can ...
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### True or false? (linear algebra) [on hold]

If $u$ and $v$ are two solutions of $Ax$ = $0$, then any vector in $Span(u, v)$ is also a solution of $Ax = 0$. I have doubts with $span(u,v)$. I do not know if the same thing $span(u\cup v)$ or ...
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### Change of (orthonormal) basis.

As I see it, the author says that $[Tv]_{e} = A[v]_{e}$ in the last paragraph. How do I see that ? I think I've jusitied the first entry in $[Tv]_{e} = A[v]_{e}$ as follows \begin{align*} \langle ...
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### $Y$ coordinate of a point that lies on a line [on hold]

Given two points $A$ and $B$, for example $A(1,5),\,B(15,2)$, what is the $y$ coordinate of a point $C(x,y)$ lying on the straight line $AB$?
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### In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
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### bilinear form and positive definiteness

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is ...
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### Efficiently solving many sets of linear equations without inversion or factorization

Suppose I have the normal set of linear equations $Ax = b$. If I can store and manipulate $A$ I have a variety of techniques available to me such as inversion, factorization, or an iterative method. ...
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### How do I find orthonormal basis of $U$?

Let $U$ be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find orthonormal basis of $U$?
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### How do I set the vector $\vec{v} = (0, 0, 1, 0, 0)^T$ as $\vec{\vec{v}}=\vec{u}+\vec{w}$ with $\vec{u}\in U$ and $\vec{w}\in U^\perp$? [on hold]

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. enter image description here
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### Can we exploit FFT for evaluating quadratic on gridded data with stationary covariance?

I would like to evaluate the quadratic $\mathbf{y}^{T}K^{-1}\mathbf{y}$ with the following assumptions: The entries of $\mathbf{y}$ are $y_i = f(\mathbf{x_i})$ which correspond to points on a ...
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### Detecting singular system during Cholesky resolution

I am solving small linear systems with a symmetric positive matrix by the method of Cholesky, without pivoting. "Bad" matrices are detected when you take the square root of a diagonal element, which ...
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### Proving Rank of a matrix is greater than its sub matrix

How can I show that the rank of a matrix is always greater than or equal to the rank of every square matrix thereof.. I mean it is self evident to anyone who knows anything about rank of matrices but ...
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### Result about Matrices of form $B(AB)^{-1}A$

I am trying to prove the following result. So far my only idea was to try using the formula for inversion of block matrices, but that did not get me very far. Any help will be much appreciated. ...
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So at the moment I'm trying to go through proofs and I came across this one: Suppose $P_n$ is the vector space of all polynomials with degree less than or equal to n. Prove that $\{1, x − 1, ... 0answers 21 views ### Proving a subgroup is a basis for a space Question: V (in R) is the subspace of all 2x2 Matrixes that are upper triangular. Prove that B is a basis for V. B= b1=$ \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ ...
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For the vectors $x$ and $y$, the Cauchy–Schwarz inequality reads $$|x\cdot y|\leq||x||\cdot||y||$$ Does this inequality only hold for 2-norm? Or for any norms? Thanks in advance.
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### Approximate spectral decomposition

See attempt below I am interested in effective and constructive computations for finding approximate spectral decompositions in some suitable format. Namely, let $A: H \rightarrow H$ be a Hermitian ...
I am trying to get a better understanding in change of basis with matrices and linear transformations, therefore I am using several linear Transformations $^{i-1}A_i=\begin{bmatrix} \cos\theta_n ... 1answer 18 views ### Assume$T$is a complex operator Assume$T$is a complex operator such that$T^{2}=T$. Prove that$Tr(T)$is a non-negative integer. There is a remark in my book, Suppose the characteristic polynomial$\chi_{T}(x)$factors intro ... 4answers 46 views ### What is a simple means of proving that 3 vectors belonging to$\Bbb{R}^2$are linearly dependent? For my linear algebra class, there is a 2 part problem that asks, given the set {(1 2), (-1 -1), (1 0)}, prove or disprove that it is linearly independent using the definition only AND then prove or ... 2answers 177 views ### Show that if A is similar to$B$and$A$is nonsingular, then$B$must also be nonsingular and$A^{-1}$and$B^{-1}$are similar I know that if B is similar to A, then B =$S^{-1}*A*S$, but I'm not sure where to go from there... 1answer 64 views ### The tangent space of a vector space I'm trying to show that there is a canonical isomorphism between a finite-dimensional vector space$V$(regarded as a$C^\infty$manifold) and its tangent space$T_vV, v\in V$, without using a basis, ... 2answers 148 views ### Is there a fundamental theorem of algebra for matrices? The fundamental theorem of algebra says we can do this ($z\in\mathbb{C}$of course) $$\sum_{k=0}^n a_kz^k= a_n\prod_{k=1}^n (z-\omega_k)=0$$ for some set$\{\omega_k \in\mathbb{C}\}_{k=1,2,\ldots , ...
From what I understand the condition number of a non-square matrix A is its largest singular value divided by its smallest nonzero singular value: $\kappa(A) = \sigma_1/\sigma_n$. Where ...