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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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Is $B = A^2 + A - 6I$ invertible when $A^2 + 2A = 3I$?

Given: $$A \in M_{nxn} (\mathbb C), \; A \neq \lambda I, \; A^2 + 2A = 3I$$ Now we define: $$B = A^2 + A - 6I$$ The question: Is $B$ inversable? Now, what I did is this: $A^2 + 2A = 3I \rightarrow \lambda^2v + 2\lambda v = 3v \rightarrow…
TheNotMe
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linear independence of functions

I am trying to prove that the set $\left\lbrace\frac{1}{n+x}\right\rbrace_{n \in \mathbb{N}}$ is linear independant in the Vector space of functions from $\mathbb{R}_{>0}$ $\to$ $\mathbb{R}$. So starting with the linear combination…
mane
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Rank after addition of positive definite matrices

I have two positive semidefinite matrices $A$ and $B$. Is it necessarily true that $$ rank(A+B) = rank(A^2+A+B) $$ ? It is easy to see that $rank(A+B) \le rank(A^2+A+B)$, but for any example I try, I end up with the ranks on both sides always…
user75267
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Find $\det X$ if $8GX=XX^T$

I need to find $\det X$ where $$8GX=XX^T,\quad G=\left(\begin{matrix}5 & 4\\3 & 2\\\end{matrix}\right).$$ My answer is that the determinant of $X$ is $-128$ and that is correct but there is one more value of $\det X$ that can solve the equation.
user1838781
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Finding the Last Eigenvalue for a Matrix

$K$ is a $3 \times 3$ real symmetric matrix such that $K = K^3$. Furthermore, we are given that: \begin{align*} K(1, 1, 1) \ \ & = \ \ (0, 0, 0) \\ K(1, 2, -3) \ \ & = \ \ (1, 2, -3) \end{align*} So we know that $0, 1$ are two of the eigenvalues of…
Andy
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Does full rank matrix have a null space?

The null space is defined as all vector that is set to null by matrix $A$, where $Ax = 0$. If the matrix $A$ is full rank, does it mean that it has no null space?
samol
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Parameters to represent degrees of freedom in $n\times n$ orthogonal real matrices

An $n\times n$ orthogonal real matrix $A$ is a set ${A_{ij}}$ of $n^2$ real numbers that satisfy the constraints: $$\sum_k A_{ik} A_{kj} = \delta_{ij} $$ for all $1\leq i,j\leq n$. The equations (1.) represent $$ n + \binom{n}{2} =…
a06e
  • 6,665
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Find matrix $B$ such that $B^2 = I - A$

$A$ is a squared matrix such that $A^4 = \mathbf{0}$. Find $B$ in form of $A$ such that $B^2 = I - A$. I tried in this way: \begin{gather*} A = I - B^2 \\ A ^ 4 = ( I - B)^4 (I + B)^4 \end{gather*} So $I = B$ or $- I = B$ But the question wants…
user767935
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For any matrix norm, is it true $||A|| \le \max|a_{ij}|\cdot ||(1)||$?

Let $|| \cdot ||$ be a matrix norm on $m \times n$ matrices, which is not assumed to be submultiplicative. Is it true that $||A|| \le \max|a_{ij}|\cdot ||(1)||$ where $(1)$ denotes the matrix with all entries equal to 1? I tried to use triangle…
Gobi
  • 7,458
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3 answers

Find a best-fit line through a point cloud that goes through a specific point.

I'm not a mathematician so I hope I ask this question properly; I apologize for anyone who is annoyed with how I ask it (I will try my best to be precise). Say I have a point cloud in $\Re^3$. I wish to fit a line through to this point cloud.…
Ben
  • 143
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affine independence

Can I, somewhat informally, state that d+1 points in $\mathbb{R}^d$ are affinely independent, if they don't lie in a $\mathbb{R}^{d-1}$ subspace?
stefan
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Calculate the orthogonal projection onto a vector $v$

I have to calculate the orthogonal projection of the vector $$ v = \begin{pmatrix} 2 \\ 4 \\ 2 \end{pmatrix} $$ on the $\text{span}(u_1,u_2)$ where $$ u_1 = \begin{pmatrix} 1/2 \\ 0 \\ 1/2 \end{pmatrix}, u_2 = \begin{pmatrix} 3/4 \\ 1/4 \\…
Mathias
  • 917
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Finding the rank of a linear transformation

I am stuck on the following problem: Let $T$ be arbitrary linear transformation from $\Bbb R^n$ to $\Bbb R^n$ which is not one-one.Then I have show that Rank $(T)=n-1.$ I know that Rank$(T)$+ Nullity $(T)=n \implies$ Rank$(T)=n-$Nullity$(T)$.…
user52976
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Kronecker delta - can I change one index and not another one in the same expression?

Ok, simple question about index notation. If I have this:- $$ \delta^\mu_\eta (\partial_\mu g_{\eta\nu}) $$ Where $\delta^\mu_\eta$ is the Kronecker delta, does this become:- $$ \partial_\eta g_{\mu\nu} $$ Or can I just switch one of the indicies…
rgvcorley
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Given $p\geq 2$ linear maps $f_i:E\to E$ such that $f_1+\ldots+f_p=id_E$ and $f_i^2=f_i,\forall i$. Prove that $f_j\circ f_i = 0,\forall i\neq j$

The full problem statement is as follows, Assume that vector space $E$ is finite dimensional, and let $f_i:E \to E$ be any $p \geq 2$ linear maps such that $f_1 + \ldots + f_p = \operatorname{id}_E$. Prove that $f_i^2 = f_i, 1 \leq i \leq p$ implies…
36cae
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