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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Derivative of a trace with respect to a matrix when the matrix is implicitly defined

I am trying to solve the following matrix maximization problem $\max_\Theta trace (A H (\Theta, P))-ln(det(H (\Theta, P)))-ln(det(P))$ , where $A, \Theta, P, F$ are all matrices and $P$ is ...
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1answer
57 views

Are symmetric binary matrices necessarily positive semi-definite?

Let $A$ be a symmetric $n\times n$ matrix with entries only 0 or 1 and the diagonal entries of $A$ are all 1. Is A positive (semi-) definite?
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1answer
27 views

Identifying Matrices of Permutations

Let $V$ be a vector space with basis $B=${$v_1,v_2,...,v_n$}. For $\pi\in S_n$ define the linear transformation $T_\pi:V \to V$ by $T(v_i)=v_{\pi(i)}$ for $1\le i \le n$. Let $M_\pi$ denote the matrix ...
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3answers
72 views

Questions regarding solvability $Ax=w$ in $\mathbb{Z}/(p-1)\mathbb{Z}$

Dear mathstack exchange, Currently I`m working on my bachelor thesis. I want to answer the following: $Ax=w$ is solveable in $\mathbb{Z}/(p-1)\mathbb{Z}$ if and only if $\mbox{det}(A) \bmod (p-1) ...
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1answer
91 views

Perspective transformation matrix application

I need to transform an angled photographed pice of paper to a "flat" image. I found this question & solution here on Mathematics and tried it out for the image given in the solution: The values ...
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2answers
484 views

How do I intuitively understand what this linear transformation matrix is?

$\begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix}$ I know how to get the product when given another matrix. But how do I know what this matrix is doing simply by looking at it?
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2answers
291 views

Can someone explain this linear transformation?

I'm having a tough time understanding how this linear transformation works...
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2answers
75 views

Prove that if $ AB^2-A$ is a non singular matrix $BA-A$ is also a non singular matrix.

Need help with this problem, tried too many times with failure. Let $A$ and $B$ be some $n\times n$ matrices. Prove that if$\space$ $ AB^2-A$ $\space$ is a non singular matrix, $\space$ ...
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1answer
59 views

the eigenvalues of matrix $X$ [closed]

Please hint me. I want to calculate the eigenvalues of matrix $ X$, which $ a,b,c,d,e$ are natural numbers. $$\mathbf{X}=\left(\begin{array}{ccc} a&b&c&d&e\\ ...
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1answer
44 views

Let p be prime. Prove that:

Let p be prime: $p^2\choose p$ is congruent to p (mod $p^2$) and $2p\choose p$ is congruent to 2(mod$p^2)$ I know that when p is prime p|$p\choose k$ where $p\choose k$ can be defined as ...
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0answers
58 views

Looking for proof verification on a linear transformation

Let $A$ be an $n\times k$ matrix, and let $B$ be a $k\times m$ matrix. $AB$ is their product $n\times m$ matrix. Prove: Part $1$: If $AB$ is one-to-one as a linear transformation, then so is $B$. ...
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25 views

If $f(x,y) = x(y^2 - x)^2 + y^5$ and $g(x,y) = y^4 + y^2 - x^2$, what would be the resultant of these two polynomials?

If $f(x,y) = x(y^2 - x)^2 + y^5$ and $g(x,y) = y^4 + y^2 - x^2$, what would be the resultant of these two polynomials? How does one compute the resultant in the first place when dealing with more then ...
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1answer
266 views

Point within a spherical triangle given areas

Consider a spherical triangle like this: where $A_1, A_2, A_3,$ and $P$ are points on the sphere and $t_1, t_2, t_3$ are the proportion of the area of the large triangle contained within the small ...
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1answer
800 views

What is Direct Sum Decomposition

Suppose that $V$ is a finite dimension inner product space and $W$ is a subspace of $V$. Then we know that $V = W \oplus W^{\perp}$. What is this $\oplus$ operator? Is it equivalent to union ...
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1answer
117 views

How would you apply the HouseHolder reflections when doing a QR factorization?

I understand the concept of using HouseHolder transformations during QR factorization, but I'm not quite sure how to actually apply them to an example. If we had some matrix, for example $$ ...
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0answers
37 views

Need help finding elementary lower triangular matricies

Given the matrix $$A= \left( \begin{array}{ccc} 1 & 2 & 1 \\ -1 & 1 & 2 \\ 2 & 2 & 4 \end{array} \right) $$ I need to find the elementary lower triangular matricies ...
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117 views

Linear independebce of $\sqrt{2}$, $\sqrt[3]{2}$, $\sqrt[4]{2}$, . . .

I was trying to prove that $\sqrt{2}, \sqrt[3]{2}, ... $ are linearly independent using elementary knowledge of rational numbers. But I could not come up with any proof using simple arguments. How to ...
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1answer
63 views

Bilinear mapping

The question is as follows, and I have added my attempt at the proof, but I don't have much information about bilinear mappings. Rudin only provided a definition. I was hoping that you all might be ...
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1answer
83 views

Vector space example

What is an example of a vector space that is a non-linear map from the real vector space of all real-valued continuous functions on R to itself?
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1answer
254 views

Matrix Linear Transformations in R3

I find this to be a very interesting problem. I extracted the vectors into a[1 5 -3] and b[2 -1 4]. For part (a), I know that the subspace is simply a space within the space R3. How would one go ...
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1answer
94 views

Is it possible to compute row and column sums of $A^{-1}$ given row and column sums of $A$?

The question is simple: We have a symmetric matrix "A" with all diagonal entries 1. Unfortunately Off-diagonal entries are unknown, but we know the row and column sum of A. Now we just need the row ...
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1answer
71 views

When (and more importantly, why) does $AA^T$ have an inverse?

Say I have a matrix A that is m by n and m < n. For context, say I am interested in finding a vector $ \theta_z \in \mathbb{R}^m$ such that: $$ A^T \theta_z = \theta_x$$ i.e. we have more ...
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2answers
55 views

Finding Range of Transformation of convergent sequences

$V$ is a vector space of all real convergent sequences. Define a transformation $T : V \rightarrow V$ s.t. if $x = \{x_n\}$ is a convergent sequence with limit $a$, $T(x) = \{y_n\}$, where $y_n = a - ...
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2answers
69 views

Prove that $\{v_1,v_2…v_n\}$ is linearly dependent if $a\notin span\{v_1,v_2,v_3…v_n\}$ and $v_n\in span\{v_1,v_2…v_{n-1},a\}$.

I ran into the next problem and got really confused: Let $\{ v_1, v_2,v_3... v_n \}$ be a set of vectors in the vector space $V$, and let $a\in V$ in such a way that: $a\notin ...
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0answers
99 views

Maximum element of Perron vector

Suppose $A$ is an entrywise nonnegative matrix that is symmetric, irreducible and has a zero diagonal. By Perron-Frobenius theorem, the spectral radius $\rho(A)$ is an eigenvalue and $A$ has, up to ...
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1answer
26 views

Given an incomplete object-object matrix containing relative size differences of the objects, how do I find the missing entries?

For example, let's say I have 3 objects $o_1, o_2, o_3$ and I am given that $o_2$ is 1 more than $o_1$, and $o_3$ is 2 less than $o_1$. I am given this information in the form of an incomplete matrix ...
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2answers
440 views

Is every invertible matrix a change of basis matrix?

In the course that I am having, we are treating change of basis matrices as the matrices of the identity operation from one basis S to another basis say B. So, our instructor introduced a theorem : ...
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1answer
199 views

Eigenvalues of special matrix with ones on the diagonals and constant $c$ on off diagonals

I came across this question in my research. If I have a p by p matrix $X$, with constant $c$ $X_{p\times p} = I_{p\times p} + c\mathbf{1}_{p\times p} - c diag(\mathbf{1})$, how do I analytically ...
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2answers
39 views

Fundamental theorem of linear algebra: The rows and columns forms a basis for a matrix $A$

In my course literature, the author claims the following: Because the column vectors of the matrix $A^T$ is equivalent to the row vectors of $A$, the following statements must be equivalent ...
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1answer
88 views

Linear Transformation Between Different Dimension Vector Spaces

If there is a linear transformation from a smaller vector space to a larger one which is 1-1 and onto (can it be)? What will happen if the transformation is from a bigger vector space to a smaller ...
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2answers
45 views

Linear Algebra - independent vectors or not

Determine if the vectors $(1,1,1,0),(0,-1,-1,1),(1,0,-1,1),(1,0,-2,1)\in \mathbb{R}^4$ are linearly independent and they can be used as a base for $\mathbb{R}^4$. How I solved this? Well, first I ...
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1answer
28 views

Dimesion of a subspace subject to linear constraints

Suppose $X$ is $n\times K$ with full column rank $K$ and $G$ is $q\times K$ with full row rank $q$. If $q<K$, how do I see that $\mathcal{L}\equiv\{Xb,b\in\mathbb{R}^K,Gb=0\}$ has dimension ...
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2answers
66 views

Problem with understanding theorem on Riccati Equation.

`The matrices $A,B,C,D,X$ are real, square, $n \times n$. I have trouble understanding theorem 7.1.2 from Lancaster & Rodman "Algebraic Riccati Equations". The part that I understand is as ...
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1answer
18 views

Relating the cardinality to surjectivity

$f(a) = (a+p\mathbb{Z},a+q\mathbb{Z}), f:\mathbb{Z} -> \mathbb{Z}/p\mathbb{Z} $ x $ \mathbb{Z}/q\mathbb{Z}$ I am trying to answer the following question on this map: I have already shown $f$ is ...
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1answer
25 views

Is $Tf = (f(-1), f(0), f(1)), T: \mathcal{C}[-1,1 ]$ surjective?

It's such a small thing, but I cant figure it out. If I have $T: \mathcal{C}[-1,1] \to \mathbb{R}^3$, given by $Tf = (f(-1), f(0), f(1))$, is that a surjective map? Proof or counterexample. Im cant ...
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1answer
18 views

How can we show the identity of the two following equations. Plane Earth Loss Model.

I would like to have a hint how to prove the following identity $\frac{P_R}{P_T} = (\frac{\lambda}{4\pi *d})^2 * | 1 + R*e^{j*k \frac{2h_Th_R}{d}}|^2$ with R = -1 this should somehow results in ...
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1answer
22 views

Norm of a Vector equality

As I prepare for the exam, I have encountered the following question: I am not very good on this Norm calculations, so your help is important. This is where I am so far: How do I continue from ...
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1answer
65 views

Largest positive eigenvalue of a matrix

I am dealing with the Capacity of constrained noiseless communication channels. It has been said that the channel capacity of such a channel is $\log{\lambda}$, which $\lambda$ is the largest positive ...
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2answers
112 views

Find the order of this matrix on the group $(GL_{2}(\mathbb{C}),\cdot)$.

I have to calculate the order of the matrix \begin{equation} A= \left( {\begin{array}{cc} i & 0\\ -2i & -i\\ \end{array} } \right) \end{equation} on $(GL_{2}(\mathbb{C}),\cdot)$. ...
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3answers
57 views

Dimension of the image of kernel of $A$ if $A^2=0$

Let A be a nonzero $3\times 3$ matrix with $A^2=0$. What is the $\dim Im(A)$ and $\dim\ker(A)$? I know $$\dim Im(A) + \dim\ker(A) = 3$$ but I don't know which is which. Any suggestions on where to ...
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1answer
59 views

Critical points of quadratic forms

Let $A$ be an $n\times n$ symmetric matrix, let $b$ be an $n$-vector, let $c \in \mathbb{R}$ and set $Q(x) = 1/2 x^T Ax-x^T b+c$. Prove that $x_0$, defined as a solution to $Ax_0=b$ is a critical ...
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1answer
75 views

Uncountable linearly independet family in $K^\mathbb{N}$

Let $K$ be a field. Consider the vector space $K^\Bbb{N}$ of $K$-sequences. Is there an uncountable linearly independent set of vectors in this vector space? If Yes, can you name it explicitely? Does ...
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1answer
79 views

How to show that if there's a fast matrix inversion algorithm, then there's a fast multiplication algorithm?

Is there a way to show this and vice versa? Suppose $F_n$ is the number of flops required by some algorithm to perform the inversion of an $n-by-n$ matrix. Assume that there exists a constant $c_1$ ...
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1answer
43 views

linear maps and ranks

Let $T : V → V$ be a linear map, where $V$ is a finite-dimensional vector space. Then $T^2$ is defined to be the composite $T\circ T$ of T with itself, and similarly $T^{i+1} = T\circ T^i$ for all $i ...
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1answer
79 views

What is the importance of last axiom of vector space [duplicate]

What is the importance of last axiom of vector space. Axiom is $1u=u$.
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187 views

What is the dimension of the set of all polynomials of degree ≤3 having a zero constant term?

I'm looking in my linear algebra book and the definition just says that dimension of the number of elements in the linear space... so how many elements are in this linear space?
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36 views

How to prove that different linear transformation on the same element will produce different elements?

How to prove that: Suppose $\sigma_1,\sigma_2\cdots,\sigma_s$are different linear transformations in linear space $V$, there must exist an element $a\in V$, s.t. $\sigma_1 a,\sigma_2 ...
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1answer
37 views

Find sufficient and necessary conditions for which the area of $A$ is zero

Let $A$ the triangle formed by the vertices $(x₁,y₁),(x₂,y₂),(x₃,y₃).$ Find sufficient and necessary conditions for which the area of $A$ is zero. If the vertices $(x₁,y₁),(x₂,y₂),(x₃,y₃)$ are ...
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0answers
112 views

If a set of $n$ vectors spans an $n$-dimensional space, it is linearly independent.

Prove that in an $n$ dimensional vector space $V$, a subset $S$ of $ V$ which has $ n$ vectors and $L(S)=V$, i.e.linear span of $S$ is $V$, is linearly independent. I can solve the problem by ...
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1answer
28 views

Transpose solution to non-homogeneous system

Let $A$ is an nxn real matrix and $A^t$ its transpose. Is it the case that if $Ax=0$ has a non-trivial solution that $A^tx=0$ has a non-trivial solution? My guess is yes, mainly because the rank of ...