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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.

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21 views

Ring homomorphisms from $\mathbb{R}$ to another unital ring $S$.

We know that there is only one non-trivial ring homomorphism from $\mathbb{Z}$ or $\mathbb{Q}$ to another unital ring $S$. What’s more,when we consider the automorphism of $\mathbb{R}$,it is unique ...
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1answer
11 views

Is there an easy way to remove scale from a squared linear transformation matrix

Given a linear transformation matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}$, I know that one can use SVD or QR decomposition ...
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0answers
17 views

LU decomposition of matrix product

Let $A_1=L_1U_1$ and $A_2=L_2U_2$ be two matrices with their respective LU-factorizations ($L_i$ is lower triangular and $U_i$ upper triangular). Is it possible to obtain the LU decomposition of the ...
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0answers
8 views

Determining the matrix in an integration by substitution context

I am working on a problem in which integration by substitution is a step. In particular, we are substituting $(u,v) \mapsto (t,z)$ s.t. $$ u = tz \\ v = (1-t)z $$ I know that conceptually when we ...
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2answers
35 views

State true or false ( I am not sure what i did wrong)

For 𝐮,𝐯 ∈ ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮+𝐯‖. The dot product of two vectors is a vector. For 𝐮,𝐯∈ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮‖+‖𝐯‖. A homogeneous system of linear equations with more equations than ...
0
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0answers
18 views

If two operators with all eigenvalues real numbers commute then both are triangularizable!

Let $A,B:E\to E$, where $E$ is a finite dimensional vector space, be two linear operators such that all of the eigenvalues are real numbers. If $AB=BA$, prove that there exists a basis in which ...
2
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0answers
11 views

Non existence of skew-symmetric matrices with the difference of their square is diagonal of some type

By using Maple I can show that there is no skew-symmetric matrices $J_1$ and $J_2$ of order 4 satisfying $$J_1^2-J_2^2=\left(\begin{array}{cccc}-\frac12Trace(J_1^2)&0&0&0\\ 0&\...
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0answers
17 views

Distance Between 2 Points in Matrix Form

Let $$ \begin{bmatrix}-m_1 &1\\-m_2 &1\\ \vdots & \vdots\\-m_n &1\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix} c_1\\ c_2\\ \vdots\\ c_n\end{bmatrix} $$ and $$ \...
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0answers
11 views

Projection on to subspace with positive constraint?

I have an underdetermined system of 2 equations in n variables and a "starting point" $s$. I need to find the point satisfying the equations which is closest to the starting point. However, I have the ...
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1answer
20 views

How to solve 3 linear equations in three variables using cross multiplication method?

How to solve 3 linear equations in 3 variables using cross multiplication method? I have no problem in solving these equations using substituting. However, how do I solve these using cross ...
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0answers
13 views

Rotation of vector by rotation matrix

Assume the following expression $$ \begin{bmatrix} a_1^* \\ a_2^* \end{bmatrix} = \begin{bmatrix} \cos(45) & - \sin(45) \\ \sin(45) & \cos(45) \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{...
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2answers
30 views

Does a right inverse of a linear map, have to be a linear map? [duplicate]

Let T be a linear map from $\mathbb{R}^3 \rightarrow \mathbb{R}^2$. Let K be a right-inverse of T. Does K have to be a linear map/transformation.
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4answers
182 views

Must every right-inverse of a linear transformation be a linear transformation?

Let T be a linear transformation $\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Let S be the right-inverse of T. Does S have to be linear transformation? Thanks in Advance.
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1answer
14 views

Can a linear transformation be expressed by constants?

Let T be a linear transformation from $\mathbb{R}^4 \rightarrow \mathbb{R}^5$. $$T \begin{pmatrix} x_1\\ x_2\\ x_3\\ x_4 \end{pmatrix} = \begin{pmatrix} x_1\\ x_2\\ x_3\\ x_4\\ 1 \end{pmatrix} $$ Is ...
0
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1answer
24 views

Projecting into a plane

Given vectors are $v_1 = (1,-2,1,3) $ , $v_2=(0,3,1,-1) $ and $v_3=(1,1,-1,-1) $ in $ R^4$ with the standard inner product. The question is how to find the orthogonal projection of $v_3$ on the ...
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0answers
12 views

Basis complement of image of sparse matrix

I have a map of vector space $R:T\to M$, and I'm interested in the quotient $E=M/R(T)$. In particular I want to construct a section $s:E\to M$. Theoretically this is not very difficult to solve, but I ...
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1answer
26 views

Given the base $B = \{(1,1,a) (0,b,-1) (c,3,0)\}$, find $a,b,c$ [on hold]

Given the base $B = \{(1,1,a) (0,b,-1) (c,3,0)\}$, find $a,b,c$ So that is that, I know that if they are a basis, then all must be independent, but at the moment I try to solve it, it gives me ...
0
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0answers
10 views

Relative position of hyper planes in $\mathbb{K}^5$ [on hold]

Considering the affine space $\mathbb{K}^5$, given the subspaces: $$X:x-1=y=z-1=t=0$$ $$X’:x=y-1=w-5=0$$ Study their relative position and calculate the set of Cartesian (implicit) equations for $X+X’...
2
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1answer
18 views

Equivalence relation on matrices

Consider $M_{n\times n}$ the $n\times n$ matrices over some field $F$. Define an equivalence relation $A\sim B$ if there is invertible $C$ such that $A=CB$. What are the equivalence classes of $\sim$ ...
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2answers
55 views

Relation between $\operatorname{trace}(AB)$ and $\operatorname{trace}(ABAB)$

Let $A$ and $B$ two square positive definite matrices of order $k$. Is there any relation between $\operatorname{trace}(AB)$ and $\operatorname{trace}(ABAB)$ for generic $k$? So far I have found that $...
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1answer
16 views

The various definitions of adjoint

Mathematics appears to have many different definitions of adjoint. There are adjoint representations on Lie algebras, adjoints of a matrix, adjoint actions on the space of linear transformations. Are ...
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1answer
28 views

Show that $\{x^4+x^3-1,x^3-x,x^2+1\}⊂E$ is a generating set of span($E$).

I'm given the following set of polynomials: $$E:=\{x^4+x^2+x,x^4+x^3-1,x^3-x,x^2+1\}$$ I know that $E$ is linearly dependent because when $\alpha_1 = -1$, $\alpha_2=1$, $\alpha_3=-1$ and $\alpha_4=1$...
3
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2answers
49 views

How to determine the base of $\ker\phi$ for polynomial function?

Given is a base defined as $$B:=(x\mapsto1,x\mapsto x,x\mapsto x^2,x\mapsto x^3 ,x\mapsto x^4)$$ A set V defined as $$V:= \{ f: \mathbb{R} \mapsto \mathbb{R}\ |\ \exists\ {a_0},...{a_4} \in \mathbb{R}\...
1
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2answers
32 views

If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$?

I'm a student and I've just read the Characteristic polynomial on Wiki. I have a feeling that: If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$ thanks to the Matrix calculator I've ...
0
votes
1answer
23 views

Finding minimum of penalty-approximated quadratic problem

Find the minimum of the following quadratic function $$ f_{\alpha}(x) := \frac{1}{2} x^T H x + c^T x + \frac{\alpha}{2}(b^Tx)^2 $$ where matrix $H$ is symmetric and positive definite, and $\...
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0answers
7 views

Computing the shifted symmetric polynomial $s^*_{(1^2)}(x_1,x_2)$

By definition, Let $\lambda=(\lambda_1,...,\lambda_n)$ be a partition with $l(\lambda)\leq n$. We define the shifted Schur polynomial in $n$ variables corresponding to $\lambda$ as \begin{equation*} ...
4
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2answers
70 views

Can a matrix $A$ commute with $e^B$ without commuting with $B$?

As in the title. Is it possible that $[A,B]\neq0$, but $[A,e^B]=0$? I tried expanding the exponential and using $[A,B^n]=\sum_k {n\choose k} B^{n-k}[A,B]B^k $ but this doesn't seem to give any insight....
0
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3answers
26 views

Prove for complex matrices $A, B$ and $C$, if $AB = BC$ than for each natural number $k$, $\\\\A^kB = BC^k $

in this question and its prove, it is stated that: For complex matrices $A, B$ and $C$, if $AB = BC$ than for each natural number $k$, $\\\\A^kB = BC^k$ I can't see why. Any help would be ...
1
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1answer
34 views

determinant of changing of basis matrix

Let $B=\{f_{1},…,f_{n},g_{1},…g_{n}\}$ the order orthonormal basis of $2n$-dimensional and $B′=\{f_{1}+g_{1},f_{1}−g_{1},…,f_{n}+g_{n},f_{n}−g_{n}\}$ the other order basis. How can I compute ...
0
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0answers
7 views

Relation between leading eigenvlaue and eigenvector of individual blocks with the leading eigenvector of non-negative symmetric block matrices

For a 2 by 2 block non-negative symmetric matrix $\mathcal{M}$, $\mathcal{M}= \left[ \begin{array}{c|c} \mathcal{A} & \mathcal{C} \\ \hline \mathcal{C}^T & \mathcal{B} \end{array} \right]$ ...
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0answers
24 views

Tangent space of matrix group

In Andrew Baker's book 'Matrix Groups', he as defined tangent space to $G$, a matrix group, at $U \in G$ as $$T_U G =\{\gamma ' (0) \in M_n(\mathbb K) : \gamma \ \text{is differentiable curve in} \ G ...
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1answer
13 views

Proving a specific $min$ function is equivalent to solving $Ax-b$

The homework question asks to prove that $min_{x\in\mathbb{R}} {f(x) = 1/2<Ax,x>-<b,x>}$ is equivalent to solving a linear system $Ax-b$. The hint the professor gave is to recite the ...
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0answers
25 views

What is the meaning of Motzkin's theorem?

Theorem: Let $A$ and $C$ be two matrices. The system of linear inequalities $Ax<0$ and $Cx \leq 0$ has a solution iff the following equation in $\lambda$ and $\mu$ does not have a solution$$A^T \...
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1answer
24 views

Set of independant vectors spanning a plane

Suppose I have a set of $n$ independent vectors in $\mathbb{R}^{m}$ where $2 < n < m$ My understanding is that these vectors do not form a basis as $n < m$, but do they span a plane in $\...
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0answers
13 views

Checking if lines intersect or skew from directions ratios and a point on each line

Well , this is what's in my book $DR_1(m1,m2,m3)$ with $A(a1,a2,a3)$ $DR_2(n1,n2,n3)$ with $B(b1,b2,b3)$ \begin{vmatrix} m1 & m2 & m3 \\ n1 & n2 & n3 \\ a_1-b_1& a_2-b_2 & ...
1
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1answer
25 views

What it means for linear functionals to be linearly independent?

Suppose we have the cannonical basis in dimension $3$, I guess three linear functionals are: $$f_1(x,y,z)=x\\f_2(x,y,z)=y\\f_3(x,y,z)=z$$ Are they linearly independent? I'm confused because if I try ...
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1answer
18 views

Find a linear map $T : \mathbb{M}_{1 \times 4}(\mathbb{R}) \rightarrow \mathbb{R}^5$ with given Kernel

Find a linear map $T : \mathbb{M}_{1 \times 4}(\mathbb{R}) \rightarrow \mathbb{R}^5$ with Kernel generated by $$ \vec{v_1} = ~[1 ~~2~~3~~4];~~ \vec{v_2} =~[0~~1~~1~~1] $$ Thoughts on this ...
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1answer
46 views

An $n \times n$ matrix $A$ is called skew-symmetric if $A^T = −A$. What values of a, b, c, and d now make the following matrix skew-symmetric? [on hold]

Let $$ A=\left( \begin{matrix} d & 8a-c & 8a+2b \\ a & 0 & 8-5d \\ a+5b & c & 0 \\ \end{matrix} \right) $$ Let $$ A^T=\left( \begin{matrix} d & ...
1
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3answers
45 views

$T : \mathbb{R}^3 \rightarrow \mathbb{R}^2$ and $S : \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Show that $S \circ T$ is not invertible.

Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^2$ and $S : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be linear transformations. Show that $S \circ T$ is not invertible. Thoughts on this problem: $S$ can ...
4
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0answers
41 views

Kronecker Product Interpretation

The algebraic expression for a Kronecker product is simple enough. Is there some way to understand what this product is? The expression for matrix-vector multiplication is easy enough to understand. ...
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2answers
40 views

Let A, B be two matrices, which satisfy Det(AB) $\neq$ Det(BA), prove rank(A) = rank(B) = 4

Let $A$ $\in$ $\mathbb{R^{4 x5}}$ and $B$ $\in$ $\mathbb{R^{5 x4}}$ which satisfy det(A$\cdot$B) $\neq$ det(B$\cdot$A). Prove rank(A) = rank(B) = $4$. So far I've tried separating the cases where one ...
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1answer
25 views

how to choose eigenvectors for repeated eigenvalue when we do spectrum decomposition?

I know that different eigenvectors from different eigenspace are automatically orthogonal. My question is: Suppose we are doing spectrum decomposition to a 3x3 symmetric matrix and we have only ...
1
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2answers
46 views

Zero of a polynomial and divisbility

I have a polynomial $p(x)$, $deg(p) = n$. I know that $\alpha$ is a zero of $p(x)$. Then $(x-\alpha)|p(x)$. Is it wrong to say that $(x-\alpha)^m|p(x)$, $m \in \mathbb{N}, m>1 $?
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0answers
18 views

Argument that the picture of P is in U

I should argument that this is $P^2 = P$ because the picture of P is in U. can I argument that if I define this vector $v ∈ R^n and \ n ∈ N$ so $v = \left(\begin{matrix} a \\b \\c \end{matrix}\...
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0answers
11 views

Derive Hessian inverse update using Sherman-Morrison in Quasi Newton Method

From Nocedal's Numerical Optimization book, the Hessian approximation equation is given using Symmetric Rank 1 (SR1) formula as follows: $$B_{k+1} = B_k + \frac{(y_k - B_ks_k)(y_k - B_ks_k)}{(y_k - ...
0
votes
1answer
19 views

Systematic approach to find base of vectorspace given its elements' traits

I'm trying to find a base for a vector space that's given as a set with certain traits. Take this example: Let $V$ be an $\mathbb{R}$-vector space with $$ V := \left\{ (a, b, c, d) \in \mathbb{R}^4 : ...
1
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3answers
30 views

How to find basis of linear subspace $V$ when $V$ contains all polynomials with the degree up to 4?

I have been given the following definition $V := \{ f : \mathbb{R} \to \mathbb{R} \mid \textrm{ there are } a_0,\ldots,a_4 \in \mathbb{R}$ and $f(x) = \sum_{i=0}^4 a_i x^i$ for all $x \in \mathbb{R}\}$...
0
votes
1answer
22 views

Questions about singular value decomposition

I'm still having trouble understanding the separation between diagonalization and singular value decomposition. I am working with complex matrices so for an arbitrary matrix, I am able to write it as $...
0
votes
2answers
40 views

If $A^n=0$, then $\left(BAB^{-1}\right)^n=0$

If $A^n=0$, then $\left(BAB^{-1}\right)^n=0$. (Assume everything that's necessary for the products to make sense.) Proof: $$\left(BAB^{-1}\right)^n=\underbrace{\left(BAB^{-1}\right)\left(BAB^{-1}\...