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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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1answer
21 views

How to transpose variables in parentheses. [on hold]

Solve for X a = b * ( c * ( X + ( d * (i * X)) + (p * X) + r + (c * X))) Can someone show me how to solve for X? Where I am so far... a = b( c( X + (d(iX)) + (pX) + r + (cX) ) ) a = b( c( X + d(...
0
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0answers
17 views

Defining Transformations given a set of elements (Apostol Volume 2)

The question is laid out like this: Let $V = \{0,1\}$ . Describe all functions $T: V\longrightarrow V$ . There are four altogether. Label them as $T_1 , T_2 , T_3, T_4$ and make a multiplication table ...
0
votes
0answers
61 views

Average angle between x and Ax?

Suppose $A\in\mathbb{R}^{n\times n}$ is a fixed positive definite matrix and $x \in \mathbb{R}^n$. The cosine of angle between $x$ and $Ax$ is given by $$ \frac{x^T Ax}{\|x\|\|Ax\|}. $$ I want to ...
0
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1answer
18 views

Markov matrices with <=1 absorbing states have all but one eigen values <1?

I found plenty of proofs online that a Markov matrix will have all its eigen values of modulus <=1. In the book on Introduction to Matrix analysis by Bellman, he also shows in section 8 of chapter ...
1
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1answer
23 views

If $A$ is a real Symmetric Matrix of order $n (\geq 2)$ , then there exists a symmetric Matrix $B$ such that $B^{2k+1} = A$.

If $A$ is a real Symmetric Matrix of order $n (\geq 2)$ , then there exists a symmetric Matrix $B$ such that $B^{2k+1} = A$. Is the statement true? I think the statement is true. My Attempt : I ...
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2answers
30 views

Proof of one of the dot product theorem

This is a statement I came up on my own and it makes intuitive sense but I'm not sure how to go about proving it. (Also I'm sure this exists already but couldn't find it online for odd reasons.) ...
0
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1answer
29 views

Free vector space of a finite set is isomorphic to its function space

Let $X$ be a finite set and let $F(X, \mathbb{C})$ be the space of functions $f: X \rightarrow \mathbb{C}$. How can I show $\mathbb{C}X$, the free vector space generated by $X$ is isomorphic to the ...
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2answers
53 views

Find k when L2 passes through the P1 with direction vector

Q: Let L1 be the line passing through the points Q1=(−3, 5, −4) and Q2=(−9, −1, 2). Find a value of k so the line L2 passing through the point P1 = P1(−1, 11, k) with direction vector →d=[1, −3, −3]...
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1answer
30 views

Doubt on solvability of AX=B

I am preparing for an exam, and I had a small doubt so I wanted to ask here. Suppose A is an $m \times n$ matrix where I have M equations in N unknowns. Consider a case where $m \lt n$ Now I will ...
1
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1answer
28 views

Rewriting the Hermitian transpose of a normal matrix as a polynomial of that matrix

In one of my tutorials today, the tutor said that he read somewhere that the Hermitian transpose of a complex matrix $A^{H}$ can always be rewritten as a polynomial of $A$, i.e. $$A^H = \sum_{i = 1}^...
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3answers
48 views

Can a matrix be not a multiple of identity, have repeated eigen values and still be diagonalizable?

The question: Diagonalisability of 2×2 matrices with repeated eigenvalues suggests that if a matrix has all its eigen values distinct, it must be diagonalizable. However, any multiple of the ...
1
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1answer
26 views

Is value in the matrix notation

Given an $m\times n$ matrix $A=\begin{bmatrix} a_{1,1}&...&a_{1,n} \\ \vdots&\ddots&\vdots \\ a_{m,1}&...&a_{m,n} \\ \end{bmatrix}$, say that I wanted to describe a set ...
1
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1answer
25 views

Show that the Image of $0$ under a Linear Mapping is also $0$

In my book of Linear Algebra, I have the following exercises: Let $T: V \to W$ be a linear map from one vector space to another. Show that $T(0) = 0$. I'm somehow having a block. For me, it is ...
1
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1answer
42 views

Dimension of a certain subspace of $M_n(F).$

Here is an interesting linear algebra problem: Let $F$ be any field and $S$ be a symmetric, invertible matrix with entries from $F.$ Then what is the dimension of the vector space: $$V = \{A\in M_n(F)...
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2answers
32 views

When does a matrix have a non-trivial solution?

Can someone please explain why this theorem is true? Theorem: If A is the matrix of coefficients of a system of linear equations, then the system has a solution if and only if the rank of the ...
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votes
1answer
40 views

How do I find an equation from vectors of a certain form?

Consider the plane $P$ in $\mathbb{R}^3$ containing all the vectors of the form: $r \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + s \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} $ Find an equation which ...
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0answers
35 views

How to compute determinant Gram matrix? [on hold]

Let $\chi_{[0,1]}$ be the Characteristic function of $A:=[0,1]\subset \mathbb R.$ Consider the set of vectors $$S=\{a_{I}:I=1 \ to \ 4\}=\{\chi_{A}(x), e^{ix}\chi_{A}(x), e^{ix}\chi_{A}(x-1), e^{ix}...
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1answer
25 views

How to identify what kind of isometry the given matrix is?

Let $E=\mathbb{R}^3$ an euclidean space and let the following matrix : $$\dfrac{1}{9}\cdot\begin{pmatrix}8&1&4\\-4&4&7\\1&8&-4\end{pmatrix}$$ Since $AA^T=I_3$ I conclude that ...
3
votes
1answer
28 views

$A^{C_p} - I_2$ has all entries divisible by $p$, for an infinite number of positive integers $p$

Let $A$ be an integer valued matrix, with $\det{A} \neq 0$. Show that there exists an infinite number of positive integers $p$ for which there exists a constant $C_p$ such that all entries of the ...
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2answers
31 views

Condition for a Linear Equation System to have non-trivial Solution

I have this Theorem in my book: For a Homogeneous System of $m$ Linear Equations in $n$ unknowns, if $m \lt n$, then the system has a non-trivial solution. I have a confusion about the condition ...
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0answers
15 views

Elements of the smallest subspace containing certain vectors

List the elements of the smallest subspace of $\mathbb{Z}^{4}_{5}$ containing the following vectors: $(0,0,0,4)^T, (2,4,3,2)^T, (1,2,4,3)^T$: Now from the definition I know that the subspace I'm ...
1
vote
3answers
54 views

Ranks of matrix

Find the rank of the following matrix $$\begin{bmatrix}1&-1&2\\2&1&3\end{bmatrix}$$ My approach: The row space exists in $R^3$ and is spanned by two vectors. Since the vectors are ...
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0answers
21 views

Derivative of a function of matrices product with respect to a single matrix

I am reading a paper and cannot understand some math that deals with a derivative of a function of matrix multiplication with respect to a single matrix. Can someone explain me how this is calculated ...
0
votes
1answer
18 views

Reference request: Convergence of singular values of tall random matrix

Let $X_n\in\mathbb{R}^{n\times m}$ be a matrix whose entries are i.i.d. random variables with zero mean and variance $\sigma^2$. Let $m$ be a fixed integer and $\|\cdot\|$ denote the 2-norm of a ...
1
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0answers
15 views

Determine dimension of set of matrices given that each matrix has same eigenvalue 0 and eigenvector

Let V be the set of all $A_{3×3}$ matrices, where all of them have eigenvalue 0 and corresponding eigenvector $y=(1,2,3)^T$. Assuming we have showed that V is a vector space, determine the dimension ...
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0answers
16 views

Frobenius norm equality related to orthogonal projections

Let $k > 1$. For which $v_1, v_2, \dots, v_k \in \mathbb{R}^{n \times 1}$ with $\Vert v_i \Vert = 1$ holds the following equation $$ \left\Vert \prod_{i=1}^k v_i v_i^T \right\Vert_F = \left\Vert \...
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0answers
14 views

Are the row vectors in a row reduced echelon matrix always independent?

Are the row vectors in a row reduced echelon matrix always independent? I'm thinking that since the first row is the only row with a non-zero coefficient, then it must be independent of all the ...
1
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1answer
32 views

Conjugate symmetry of an inner product

I want to prove the following: $\langle A,B \rangle = \overline{\langle B,A \rangle}$ where $\langle A,B \rangle := tr(AB^{*})\,\, and \,\, A,B \in \mathbb{C}^{n \times n} $ Note: The bar ...
1
vote
1answer
61 views
+50

Showing equivalence of set of solutions of two linear programming

Setup of the problem (updated thanks to comments below) Consider six finite sets of real numbers each with cardinality $4$ $$ \mathcal{A}\equiv \{a_1,a_2,a_3,a_4\}\text{, }\text{ }\tilde{\mathcal{A}}\...
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1answer
26 views

Prove that basis of sum of subspaces contains bases of subspaces and the intersection

We have a vector space $V$ and two subspaces $S$, and $T$. We know that $\{x_1, x_2, ..., x_k\}$ is a basis of $S\cap T$, such that $\{x_1, x_2, ..., x_k, y_{k+1}, ..., y_n\}$ is a basis of $S$, and $\...
2
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0answers
22 views

Orthogonal matrices and linear isometries

I've been stuck trying to prove this theorem for a while, so I decided to ask for someone's help here. The theorem goes as follows: "let $f$ be a linear isometry of some Euclidean vector space; then ...
1
vote
2answers
28 views

Find a determinant for a linear map $Z \in M_{7×7}(\mathbb R)$ if you know that $Z^{2}-8Z^{-1}$ is a zero matrix

I have a trouble with this task because I think that I need clever way to do this easy and fast. However I don't have any idea to don't count it. My only idea is to firstly calculate $Z^{2}$ but then ...
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0answers
17 views

Adjoint operator on $\mathbb{L} ^2 _\mathbb{R}$ of $\frac{d}{dx}$

If A is a linear operator defined on $\mathbb{L} ^2 _\mathbb{R}$ of $\frac{d}{dx}$ then its adjoint must satisfy the property: $$(A^*f,g) = (f,Ag) \\ f,g \in \mathbb{L} ^2 _\mathbb{R}$$$ Now if $A = \...
2
votes
1answer
25 views

How can I continue the solution? Find the Kernel of a vector space.

We are given the vector space $V=F^3$ where $F$ is a field and we are given $B=\{v_1,v_2,v_3\}$ is a basis in $V$, and $T: V \rightarrow V$ is a linear transformation. We are also given: $$[T]_B=\...
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0answers
21 views

an example of a non-linear isometry between normed spaces which is not one-to-one

Can anybody give an example of a non-linear isometry between normed spaces which is not one-to-one? Or, does there exist such a mapping?
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0answers
14 views

Given $A \in SL_n(\Bbb R)$ Find $B,P \in SL_n(\Bbb R)$ such that, $A=BPB^{-1}P^{-1}$

I was trying to prove that, $[SL_n(\Bbb R), SL_n(\Bbb R)]=SL_n(\Bbb R)$ My attempt: $[SL_n(\Bbb R), SL_n(\Bbb R)]\subset SL_n(\Bbb R)$ is trivial, trying to prove the other inclusion. So it's ...
2
votes
3answers
68 views

Minimum of $||\mathbf{x-y}||^2+||\mathbf{Ax}-\mathbf{b}||^2$

Given a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ and vectors $\mathbf{b}\in\mathbb{R}^m$ and $\mathbf{y}\in\mathbb{R}^n$, I need to find the vector $\mathbf{x_*}\in\mathbb{R}^n$ such that $$\...
2
votes
2answers
49 views

Solve matrix-2 norm problem with diagonal matrix constraint.

How does one solve the following problem (matrix-2 norm and diagonal matrix constraint) analytically: $$\hat b = \arg \min_{b} f \left( b \right)$$ such that $$f \left( b \right) = \left\|A- {F}^{...
0
votes
1answer
36 views

Calculate the distance of the point p from the affine hyperplane H

I am doing a linear-algebra course and the question I get is what the distance is from any point p to an affine hyperplane H in $\mathbb{R}^3$ given by: $\begin{equation} H=\{\,h0+\alpha h1+\beta h2\,...
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0answers
15 views

Show that $A^o$ is convex, balanced, closed in $A'$

Let $E$ is a normed space and $A \subset E$. Define $$A^o := \{y \in A': |y(x)| \leq 1, \forall x \in A\}.$$ in which, $A'$ is the dual space of $A$. With this definition, how to show that $A^o$ is ...
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0answers
17 views

quadratic eigenvalue problem reformulation with eigenvalues on unit cycle

Suppose a quadratic eigenvalue problem is $$(A_2\lambda^2+A_1\lambda+A_0)x=0$$ where $\lambda$ is the eigenvalue and $x$ is the eigenvector. the quadratic form can be rewritten by $$\left[ {\begin{...
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0answers
19 views

What is the complete definition of inner product space

There are some contradicting and confusing definitions online and I am trying to figure out the general case definition. For example it is seems that order between elements in the range of the inner ...
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0answers
27 views

general linear group over a vector space

Let $V$ be a vectors space with given basis, say $a_1, a_2, \ldots a_n$ and let $\mathbb{L}$ be an algebraic closure of a field $\mathbb{K}$. What should $GL(V_{\mathbb{L}})$ mean? and also, up to ...
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1answer
41 views

Can we simplify $ A^{-1}Bx = x$ where $A$ is a block matrix with each block being diagonal and half the blocks of $B$ are zero?

I have the following eigenvalue problem involving block matrices $A$ and $B$: $$ A^{-1}Bx = x. \quad \quad \quad \quad (*) $$ $A$ and $B$ have special structures. I would like to reduce/simplify this ...
2
votes
1answer
358 views

Prove that the following determinant equals $0$

We have a $n\times n$ matrix $A=(a_{i,j})$. If $i=j$, then $a_{i,j}=1-n$. otherwise, $a_{i,j}=1$. Show that $|A|=0$. I tried using gauss elimination but it just gets too complicated. I also tried to ...
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1answer
72 views

Prove that $x^{m+n}=x^mx^n,x\in G$ group,$\forall m,n\in\mathbb{Z}$ [on hold]

The part that I did not fully understand is marked in yellow Definitions: $(A,\cdot)$ Magma $1\in A$ neutral element, $x\in A$ $x^0:= 1, \forall_{n\in\mathbb{N}_0} x^{n+1}:=x\cdot x^n$ $(1)$ $(...
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vote
3answers
22 views

Study the injectivity and surjectivity of the function $D:\mathbb{R}_n[x]->\mathbb{R}_{n-1}[x], D(f) = f'$

As the title says, I want to study the injectivity and surjectivity of the function: $D:\mathbb{R}_n[x]->\mathbb{R}_{n-1}[x], D(f) = f'$ I think it is pretty straight forward that the function is ...
3
votes
0answers
51 views

What is the number of elements in $\{A=(a_{ij}) \in Gl_n(\mathbb{F}_q):a_{11}=1\}$

Let $\mathbb{F}_q$ be the finite field with $q$ elements. I want to calculate the number of elements of the set $S=\{A=(a_{ij}) \in Gl_n(\mathbb{F}_q):a_{11}=1\}.$ We know that $|Gl_n(\mathbb{F}_q)|=(...
0
votes
2answers
96 views

Is $a=b\Rightarrow a+c=b+c$ always true?

Let $(A,+)$ be a Magma that means it is not a $field$ is the above identity still true? I have encountered the Substitution law at my first proof that if $A$ is a Group then ist Right neutrall ...
2
votes
1answer
21 views

Similarity of two tridiagonal matrices

I am considering two complex symmetric tridiagonal matrices. First, A is a tridiagonal matrix with identical non-diagonal elements : A = $\begin{pmatrix} ig_1 & \kappa & 0 & 0 \\ \kappa &...