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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1answer
32 views

Characterising the cone of elements whose inner product with $v \otimes v$ is non-negative

Let $V$ be an $n$-dimensional real inner product space. Consider the space $V \otimes V$, endowed with the tensor product metric, i.e. $$ \langle v_1 \otimes v_2 , w_1 \otimes w_2\rangle := \langle ...
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29 views

Comparing condition numbers

I have two matrices, $A$ and $B$ where $A,B \in R_{mxn}$ ($m>>n$). I need to answer the question: Which one is better-conditioned among $A$ and $B$ ? However, I will be making these comparisons ...
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1answer
10 views

start T in matrix equation

In many start equations I see T (transpose) or -1 (inverse). Why is there using T, but not original matrix? Example https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula ($\mathbf{A}$ + $\...
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2answers
39 views

Similarity transformation into symmetric matrix

I have a matrix of the form: $$ \begin{bmatrix} 0 & q & 0 & 0 & 0 & 0 & \cdots \\ p & 0 & q & 0 & 0 & 0 &...
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0answers
25 views

The solution space of $A_{12}X=0$ and $B_{12}X=0$ is isomorphic?

Let $A$ be a $n\times n$ invertible matrix. Suppose $$A=\begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22} \end{pmatrix}$$ $$A^{-1}=\begin{pmatrix}B_{11}&B_{12}\\ B_{21}&B_{22} \end{pmatrix}...
1
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1answer
24 views

Linear transformation diagonalization with unknown vectors in basis

I've been working for some hours with this problem but I still can't get it. The problem says as follows: Given $ B=\{V_1, V_2, V_3\} $ and $ B'= \{V_1, V_1+V_2,-V_1-2V_2-V_3\} $ , basis of a vector ...
0
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2answers
71 views

Do real eigenvalues $\implies$ symmetric matrix? And why is a positive definite matrix symmetric?

Proof: $Av$ = $\lambda v$ $\implies \bar{v}^{T}Av = \lambda \bar{v}^{T} v$ ------(1) And, $Av$ = $\lambda v \implies \bar{A}\bar{v}$=$\bar{\lambda}\bar{v} \implies \bar{v}^{T}\bar{A}^{T}=\bar{\...
1
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1answer
33 views

Question about points in general position (from Keith Conrad's expository paper on isometries)

https://kconrad.math.uconn.edu/blurbs/grouptheory/isometryRn.pdf In an expository paper on isometries of $\Bbb R^n$ Keith Conrad proves the following corollary: Corollary 2.7. Let $P_0,...,P_n$ be ...
3
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1answer
51 views

Showing that there is unique matrix $B$ such that $B^k=A$ for some $A$

Let $A$ be a $n$ by $n$ real matrix with distinct positive eigenvalues $\lambda_1$,...,$\lambda_n$. And let $k$ be an odd integer. Then, I was able to show that there exists a real matrix $B$ such ...
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1answer
33 views

For $4\times 3$ matrix $M$, for any $3\times4$ matrix $N$, $\exists 0 \neq v \in \mathbb{C}^4$ such that $MNv = 0$.

Define $M := \left(\begin{matrix}1&-1&2\\2&-1&1\\-4&1&0\\3&-2&3\end{matrix}\right)$. Prove or disprove: For every $3 \times 4$ complex matrix $N$, there is a non-zero ...
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0answers
21 views

Prove that if a linear system has infinitely many solutions, then any solution could be written as a linear function of free parameters, thanks.

Suppose $Ax=b$, where $A$ is of dimension $q\times p$ with $q<p$, $rank(A)=q$, $b$ is $q \times 1$. Let $\mathcal{X}=\{x:Ax=b\}$ be the solution set of this system. How to rigorously prove that ...
2
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3answers
42 views

$V=\{A\in M_n(\mathbb Q): \operatorname{tr}A=0\}$, Prove that $V\oplus \operatorname{Span}\{I_n\}=M_n(\mathbb Q)$

Q: Let $V=\{A\in M_n(\mathbb Q): \operatorname{tr}A=0\}$. Prove that $V\oplus \operatorname{Span}\{I_n\}=M_n(\mathbb Q)$ Since $\dim(V\cap \operatorname{Span}\{I_n\})=0$, $\dim(\operatorname{Span}\{...
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0answers
26 views

A polynomial basis of Cn[X]

Let $\mathrm{c}_{0}, \ldots, c_{n}$ be pairwise disctinct complexes. I proved that $\left(\left(X-c_{i}\right)^{n}\right)_{0 \leq i \leq n}$ is a basis of $\mathbb{C}_{n}[X]$ by induction. 1) Do you ...
3
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1answer
51 views

Why are self-adjoint operators important?

I am learning about self-adjoint and normal operators. So far, they have come up in the Spectral theorem, which says self-adjoint operators have an eigenvalue basis and a corresponding diagonal ...
1
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2answers
49 views

Inductively simplify specific Vandermonde determinant

From Serge Lang's Linear Algebra: Let $x_1$, $x_2$, $x_3$ be numbers. Show that: $$\begin{vmatrix} 1 & x_1 & x_1^2\\ 1 &x_2 & x_2^2\\ 1 & x_3 & x_3^2 \end{vmatrix}=(...
1
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1answer
43 views

Prove: if $W$ is subspace of $V$ and vector $v \in W$, then orthogonal projection of vector $v$ onto $W$ is $v$ itself

Let's assume that $\bar v \in V$. Let's also assume that $\bar v \in W$, where $W$ is a subspace of $V$. How to prove that then orthogonal projection $\operatorname{proj}_{W} (\bar v) = {\left\...
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0answers
22 views

Problem on quadratic form

Show that $\{x : x^{T}Ax \le 1\}$ is bounded if $A$ is positive definite, where $x^T=[x_1,x_2,...,x_n]$ and $A$ is an $n\times n$ matrix.
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2answers
64 views

when is $AB = B^TA$ true?

In what particular situation would the following be true? $AB = B^TA$ where $A$ is symmetrical, $B$ is not. I also know that $BB = B$.
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0answers
47 views

Writing the dual of a linear minimisation problem

I am trying to write down the dual of a linear minimisation problem and I would like your help to double check whether I'm doing it right. I'm following the instruction here . The original ...
4
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1answer
33 views

Generalizing the conjugate gradient like this works?

Given $A \in \mathbb{R}^{n \times n}$, a SPD matrix, and a vector $b \in \mathbb{R}^n$, it is possible to solve the problem $$\min_x \| Ax - b\|$$ with the conjugate gradient method. Its algorithm ...
0
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1answer
26 views

Let $U$ be a subspace of $V$, show that $(U^{0})_0 = U$

It is pretty obvious why $(U^{0})_0 \supseteq U$. but how do I show that $(U^{0})_0 \subseteq U$? For clarification: $U^{0} = \left \{ \phi\in V^{*} \mid \phi(a) = 0, for \; u \in U\right \}$ $(...
0
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1answer
32 views

If $V$ is a vector space, $W$ and $U$ are subspaces of $V$, why is $U^0, W^0 \supseteq ( U \cap W)^0$ true?

If $\textsf V$ is a vector space, $\textsf W$ and $\textsf U$ are subspaces of $\textsf V$, why is $\textsf U^0, \textsf W^0 \supseteq (\textsf U \cap \textsf W)^0$ true? My mind tells me that the ...
2
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1answer
20 views

Dimension of the space of matrices which is commutative to a given matrix.

Suppose I have a matrix $A $ in the space $ V $ of $n $ by $n $ matrices. Then it is quite clear that $S=\{B : AB=BA\} $ form a subspace. I want to find out its dimension. I think it depends on the ...
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0answers
25 views

Proof that equation $Lf=u$ guarantees satisfiability of weak form $\langle Lf,v \rangle = \langle u ,v \rangle$

Problem Let's observe in a closed interval $[a,b] \subset \mathbb{R}$ real-valued and continuous vectorspace $\mathcal{F}([a,b],\mathbb{R})$. Where $\langle ., .\rangle$ is some scalar product. This ...
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1answer
48 views

A quadratic in two variables can be factored if the determinant is $0$

(All real numbers) Show that $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ can be factored as $$(a_1x+b_1y+c_1)(a_2x+b_2y+c_2)=0$$ iff $$\begin{vmatrix} a&h&g\\ h&b&f\\g&f&c \end{vmatrix} = ...
1
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1answer
21 views

Linearly dependent matrix columns implies zeroes in the solution of the system of equation given by this matrix?

Let's assume that we have a system of equations: $a_{11} x_1 + ... + a_{1n} x_n = 0$ $a_{21} x_1 + ... + a_{2n} x_n = 0$ $......................$ $a_{m1} x_1 + ... + a_{mn} x_n = 0$ Let us denote by ...
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0answers
31 views

Doubt in Hoffman and Kunze Section 5.2 (existence of determinant)

I am trying to read Hoffman Kunze's book on linear algebra and I have a doubt in a particular result, (Theorem 1) of Section 5.2. Specifically, the theorem states: Let $n > 1$ and let $D$ be an ...
2
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0answers
25 views

Different approaches to calculating angle between two vectors

I can calculate angle θ between two dimensional vectors a and b as the inverse cosine of their dot product divided by their magnitude: ...
1
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1answer
32 views

Product of two matrices in block form: step in proof

Let $R$ be a commutative ring with $1$ and $M_{k,l}(R)$ denote set of matrices of size $k\times l$ over $R$. $A\in M_{m,n}(R)$ and $B\in M_{n,p}(R)$. Partition $m$, $n$ and $p$ as $$ m=m_1+\cdots + ...
0
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1answer
29 views

To prove a matrix is PSD

This question rises from the proof of the outer product Cholesky Factorization. If the matrix $$ M=\begin{pmatrix} \alpha&\vec{q}^T \\ \vec{q}&N \end{pmatrix} $$ is positive semidefinite with ...
0
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0answers
38 views

Quaternion Matrix

Define $$ 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ $$ i = \begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix} $$ $$ j = \begin{...
5
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0answers
59 views

If $\mathbf{R}$ is an upper triangular matrix, then does $\|\mathbf{R}\| \le \|\mathbf{R} + \mathbf{R}^T\|$ hold?

Let $\mathbf{R}\in\mathbb{R}^{n\times n}$ be upper triangular and $\|\cdot\|$ be the induced 2-norm of matrices. Then, does $\|\mathbf{R}\| \le \|\mathbf{R} + \mathbf{R}^T\|$ hold?
3
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2answers
54 views

Show that $A$ forms a basis of the span of $A$.

Suppose that $A=\{y_1,...,y_r\}$ is a subset of a vector space $V$ and that every vector $x \in V$ can be expressed uniquely as a linear combination of the vectors of $A$. Show that $A$ forms a basis ...
0
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1answer
37 views

matrix and vector derivative

Suppose $~W~$ is an $~n \times n~$ matrix and $~x$,$~y~$ are $~n \times 1~$ vectors. Then define function $~f~$ as following: \begin{equation} f=\left\|W\left(x-y\right)\right\|^{2} \end{equation} ...
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0answers
72 views

Is it possible to find the explicit inverse of $I+AB$, where $A$ is diagonal and $B$ symmetric matrix?

Question. Is it possible to find the explicit inverse of $I+AB$, where $A$ is diagonal and $B$ symmetric matrix? $A$, $B$ are non-singular, and $I+AB$ is invertible. For $B=S-I$, where $S=[s_{ij}]=[c*...
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0answers
36 views

Advise courses on linear algebra and multivariable calculus? [on hold]

Required courses: linear algebra, multivariable calculus. More precisely, in the future I want to try to go to machine learning and I looked at what is needed for this. Multivariable calculus, linear ...
-1
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1answer
21 views

Minimum value of a Given function and the triplet coordinates. [on hold]

How to solve this question? Question Let a be a positive real number,and define the function f(x,y,z)of real variables by the quadratic form \begin{equation*} f( x,y,z) =x^{2} +y^{2} +z^{2} +2ayz \...
1
vote
2answers
30 views

Volume of a frustum given the bottom radius and the top cone height.

A cone with base radius 12 cm is sliced parallel to its base, as shown, to remove a smaller cone of height 15 cm. If the height of the smaller cone is three-fourths that of the original cone, what ...
0
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1answer
61 views

Why does $(A^TA)^{-1} = I$ imply that $T_A$ is injective?

Let $T_A : \mathbb R^n \to \mathbb R^m$ be given by the matrix $\mathbf{A}$. I have been told A has a left inverse if $T_A$ is injective. Also, I have been told that $\mathbf{A}$ has a left inverse if ...
0
votes
1answer
19 views

Mathematical notation for orthogonal projection of a set of points on a line

How could i mathematically denote the following: Assume a matrix P which represents the coordinates of a set of points (each row = a single point). Each point in the matrix P is projected on a line ...
0
votes
1answer
23 views

Collinear vectors, simulation

When I draw two collinear vectors, say $a(2,4,6)$ and $b(4,8,12)$, using a simulator, they end up being the same line and starting from the origin. They are not separated in space. The usual images of ...
1
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1answer
36 views

Solving a linear equation in one component of $\Bbb{Z}^3$.

Consider the space $X = \Bbb{Z}^3$, a $\Bbb{Z}$-module. Let $M = \{ \sum_{i=1}^n c_i(p_i, q_i, r_i) : \sum_{i=1}^n c_i q_i = 0,$ where $p_i, q_i, r_i$ are either prime numbers or $0 \}$. Then is $M ...
0
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2answers
48 views

Example of $2\times 2$ matrices with common eigenvalues

Give an example of $2\times 2$ matrices $\mathbf{A, B}$ such that $\forall t \in \mathbb{R}$ the matrix $\mathbf{A} + t\mathbf{B}$ has the eigenvalues $\pm\sqrt{t}$. I presume that there are no ...
0
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1answer
26 views

A problem on characteristic values.

If the characteristic values of $\begin{pmatrix} 3 & -1 \\ 5 & 6 \end{pmatrix}$ are $a$ and $b$. And of $\begin{pmatrix} 1 & 2 \\ -1 & 5 \end{pmatrix}$ Are $c$ and $d$...
2
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1answer
72 views

Basis for $\operatorname{null}T$ given basis for $\operatorname{range}T$

I was trying to solve exercise 24 of chapter 3.B of "Linear Algebra done right", by Sheldon Axler. It states: Suppose $W$ is finite-dimensional and $T_1, T_2 \in \mathcal{L}(V, W)$. Prove that $\...
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0answers
33 views

Orthogonal Projection in Subespace Proof

Let $M$ be a subspace of $R^n$ and $z\notin M$. Show that the orthogonal proyection of $z$ in $M$ is $\bar x$ if and only if: $(z-\bar x,x)=0, $ $ \forall x \in M$ How can i prove it? I know that ...
0
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2answers
26 views

How to find one-sided inverse of a non-invertible linear transformation?

Suppose I am working with the linear transformation from $\mathbb R^3$ to $\mathbb R^2$ given by a $2\times3$ matrix say $$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 0 & 5 \\ ...
1
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1answer
36 views

Representing a Bilinear Form in a Matrix

Let $b : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form, and $\langle,\rangle$ be the standard inner product of the Euclidean space and $e_1,...e_n$ be the standard basis for the ...
1
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2answers
38 views

Prove the existence of left and right inverse?

A theorem states the following Let $\mathbf{A}$ be a $m \times n$ matrix such that rank $\mathbf{A} = r$. (a) $\mathbf{A}$ has a right-inverse if and only if and only if $r = m$ and $m \leq n$ (b) $...