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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Bilinear form vs two-form

I have a basic linear algebra question since I'm confused with the definitions: What is the difference between bilinear form and 2-form? I looked up in Wikipedia and it says that a linear form $B(v,...
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25 views

Finding the limit of a Matrices determinant

The problem is as follows: I've been trying to figure this out with no luck. I'm lost at the $A_k+1$ and $A_0$. I'm not sure what they are implying and how they would apply in finding the limit.
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1answer
28 views

Vector as a linear combination of other Vectors

I have a set of vectors and in $\mathbb R^3$ in a $3\times 6$ matrix already in row-reduced echelon form. I had to express one of $v_1(1, 0, 0), v_3(0,1,0)$, and $v_6(3,-2, 0)$ as a linear combination ...
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23 views

How can I solve for k for this linear transformation?

A linear transformation T: $\mathbb R^3 \rightarrow \mathbb R^2$ whose matrix is $\begin{bmatrix}-3&-9&-9\\-3&-9& -10+k\end{bmatrix}$ is onto if $k \ne ? $ I put this matrix in ...
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1answer
45 views

Matrix Trace Inequality [on hold]

If $\operatorname{Tr}(A) < \operatorname{Tr}(B)$, is it fair to say that $\operatorname{Tr}(AC) < \operatorname{Tr}(BC)$? All of $A$, $B$ and $C$ are positive definite matrices.
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Define the linear transformation T

Let $A = \begin{bmatrix}4&-5\\-3&-2\\23&-23\end{bmatrix}$ and $b = \begin{bmatrix}-37\\-1\\-184\end{bmatrix}$ Define the linear transformation $T: \mathbb R^2 \rightarrow \mathbb R^3 |...
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1answer
46 views

Eigenvalues and eigenvectors of $I \otimes A \ + \ B^T \otimes I$ (used in Sylvester's equation)

Let $A$ and $B$ be $n \times n$ square matrices, with resp. eigenpairs $(\lambda_i,U_i)$ and $(\mu_j,V_j)$. Let $I_n$ be the order $n$ identity matrix. I have seen a result that says that the $n^2$...
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65 views

Prove that N(T)=0 and R(S)=U

Let $T:U \to V$; $S:V \to U$ and $ST:U \to U$. Prove that $N(T)=\{0\}$ and $R(S)=U$. My professor gave us a fact at some point that if $ST=ID(U)$ we have S is surjective and T is injective. I am not ...
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26 views

Finding a pair of vectors in $u$, $v$ in R that span the set of all x in R^4 that are mapped into the zero vector.. [on hold]

Find a pair of vectors in $u$, $v$ in $\mathbb R^4$ that span the set of all $x \in \mathbb R^4$ that are mapped into the zero vector by the transformation $x\rightarrow Ax$ Let $A = \begin{...
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73 views

Prove that $\exp(A+B)=\exp(A)\exp(B)$ iff $[A,B] = 0$

I have searched throughout the forum and online as well, and I got that with condition of $[A,B]=0$, $e^{(A+B)t}=e^{At}e^{Bt}$. Now the question is, to show for any matrices $A$ and $B$, it is true ...
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49 views

Possible Values of $\dim(Null(L))$

can you please help me check this question down vote favorite can you please explain this question to me? Thanks Question : Suppose that $L: \Bbb{R}^4 \to \Bbb{R}^2$ is a linear transformation. a)...
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0answers
23 views

Step in derivation of symmetric random matrices spectrum

I was trying to go through a paper about 'The eigenvalue spectrum of a large symmetric random matrix' by Edwards and Jones (1976) and I found myself stuck at the very first step of a derivation. I ...
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1answer
32 views

What is the intuition behind Gramian method for linear independence? and Is there $simple$ proof of it?

I'm trying to figure out the intuition behind Gramian method to determine the linear independence of functions. I searched the web for such simple intuitive explanation and found nothing. I tried ...
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4answers
75 views

prove triangular inequality for $ d(x,y)= \frac{||x-y||}{1+||x-y||}$ [duplicate]

prove triangular inequality for $$ d(x,y)= \frac{||x-y||}{1+||x-y||}$$ that is $d(x,y) \leq d(x,z)+d(z,y)$ ofcourse ||.|| is a norm and has properties of norms this usually works $$ \begin{...
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26 views

Convex combination of projection operators

If $P_1, P_2: V \to V$ are linear projection operators on the vector space $V$ with $R := P_1(V) = P_2(V)$, is it true that any convex combination of $P_1$ and $P_2$ is again a projection operator $...
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37 views

Does encountering a zero pivot during Gaussian elimination imply that the matrix is singular?

I was reading a problem about Gaussian elimination and pivots of a matrix, say $A$. The question is: During the Gaussian elimination process without pivoting a zero pivot has been encountered. Is ...
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35 views

Can we define component of a matrix which is orthogonal to another matrix?

Given two vectors $A$ and $B$ one can easily find component of $A$ along $B$ and component of $A$ perpendicular/orthogonal to $B$ and vice versa. This is possible as we can define dot product of two ...
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3answers
95 views

For which $a$ and $b$ is this matrix diagonalizable?

For which $a$ and $b$ is this matrix diagonalizable? $$A=\begin{pmatrix} a & 0 & b \\ 0 & b & 0 \\ b & 0 & a \end{pmatrix}$$ How to get those $a$ and $b$? I calculated ...
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3answers
38 views

Is the Gramian determinant always nonnegative?

Is Gramian determinant $\det (A^TA)$ always nonnegative (or at least when $A$ has no more columns than rows)? It's used to compute a volume element as in this article https://en.wikipedia.org/wiki/...
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For an nxm matrix A, what is dim(im(A)) + dim(ker(A^T))?

So I know that the rank nullity theorem says that $dim(im(A)) + dim(ker(A))$ is the number of column in the matrix, and that seems like it would be useful, but here we have $A^T$ instead of A, so I'm ...
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24 views

How to find the angle between two tables given 2 points on table 2 and origin on table 1?

Question How to find the angle between two tables of same dimensions when 2 points on the second table is known with respect to the origin(0,0,0) on the middle of the first table? Knowns 1.origin on ...
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81 views

Prove that the number: $z = \det(A+B) \det(\overline A-\overline B)$ is purely imaginary.

Problem: Let $A_{n\times n}$ and $B_{n\times n}$ be complex unitary matrices, where n is an odd number. Prove that the number: $$z=\det(A+B) \det(\overline A-\overline B)$$ is purely imaginary. My ...
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Polar set of orthogonal matrices set is nuclear norm ball

Reltated problems: Show that the dual norm of spectral norm is Nuclear norm. Proof that nuclear norm is convex. The set of orthogonal matrices is defined as: $$\mathcal{O}(n) = \{X\in \...
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1answer
94 views

Seeking 2 values from a 3 value equation [on hold]

I have been searching for a compression method that can be expressed as a math formula. I decided that since $0-127$ is the range, and $3$ is a nominal prime number, that I would perhaps get some ...
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3answers
129 views

A certain unique rotation matrix

One can find that the matrix $A=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & -\dfrac{1}{3} & \dfrac{2}{3} \\ \...
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1answer
60 views

problem in linear algebra is unsolved though it is very fundamental ?help

Let A be a $ n\times m $ matrix and b be a $n \times 1$ vector(with real entries). Suppose the equation $Ax=b,x\in R^m$ admits a unique solution . Then whether $m=n$ or $m\leq n$ . I know that if n=m ...
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109 views

Show that an integer matrix with following conditions is the identity $I$

every entries of $A$ is integer every entries of $A-I$ is multiple of a prime $p$ ($p\geq3$) there exists $n\ge1$ such that $A^n=I$ show that $A=I$ I tried $A=I+p^kB$ where not every entries of $B$ ...
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Find and orthogonal basis of $R^3$ such that $M(\phi,C)=A$

Considering the matrix $$A=\begin{bmatrix}{0}&{2}&{-2}\\{2}&{0}&{1}\\{-2}&{1}&{0}\end{bmatrix}\in M_3(R).$$ Find an orthogonal basis for the bilinear form of $R^3$ such ...
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1answer
39 views

Quadratic Forms Using Derivatives

This link says we can diagonalize a quadratic form $$ f(\vec{x}) = \sum_{i,j=1}^n a_{ij}x_i x_j, $$ $$a_{ij} = a_{ji}, a_{ii} \neq 0$$ using derivatives (?!!!) in a formula like $$f(\vec{x}) = \...
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A few questions about linear maps [on hold]

I have a few problems with linear maps. Any help would be appreciated First one: if I'm given the associated matrix of a linear map on the canonical basis and another matrix of the same map on a ...
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1answer
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Trying to find 2 equalities

STRICTLY: I do not need coding help. I just need to know why 2 equations won't come to a single equality I'm trying to use a conditional statement made of 4 equations to see if it is possible to take ...
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2answers
45 views

Solving Volume with area only given

Just some quick help I want to know if I did this correctly because I don't have an answer sheet. Question: If $2700cm^2$ of material is available to make a box with a square base and open top, find ...
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1answer
46 views

How to find the matrix base change in this vector space?

Good Morning. I'm trying to solve this exercise: I took the quotient vector space formed by $\operatorname V=\mathbb{R}^5/\langle(3,2,4,-2,5)\rangle$. After starting the vector $x=(3,2,4,-2,5)$ I ...
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1answer
36 views

Inverse of a matrix with main diagonal elements approaching infinity

Let $A$ be a invertible, symmetric, positive definite $p \times p$ covariance matrix with main diagonal elements $a_{ii},~i = 1,~\ldots,~p$. If all main diagonal elements would approach $\infty$, ...
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1answer
31 views

Sum of orthogonal complements of two subspaces

If $V_1$ and $V_2$ are subspaces of vector space $V$, where $V$ is a finite-dimensional inner product space then: $(V_1+V_2)^⊥=V_1^⊥+V_2^⊥$ So far I have tried showing this, by taking a vector $u \...
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3answers
68 views

Uniqueness of solution for a tridiagonal system

I have a claim I've been conjecturing. Not sure if it's true or not. Context: I'm doing some calculations with finite difference schemes. Say I have the following real $n$ x $n$ tridiagonal matrix $A$...
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1answer
30 views

Show that following three statements are equivalent

Proposition: Let $V$ be finite-dimensional inner-product space and $A\in L(V)$. Show that following three statements are equivalent:1) A is hermitian operator.2) For every orthonormal basis $b$ in $V$ ...
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3answers
43 views

Basis for a subspace

I need to calculate the basis for $$W = \lbrace (a,b,c,d) \: : \: a+b+c = 0 \rbrace.$$ I find it hard to understand how does the fact that d is not part of the equation effects the basis. Thanks ...
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1answer
32 views

Prove a specific property for tetrahedron

I have the following question. If the heights from vertices $A$ and $D$ in tetrahedron $ABCD$ intersect then $AD$ and $BC$ are perpendicular. I draw a sketch of the tetrahedron but I don't have any ...
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1answer
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Factorising using the factor theorem

I'm getting an early start on practicing for the GRE. I'm trying some hard questions because I haven't done maths since high school and need to challenge myself. I'm looking at factor theorem right ...
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1answer
22 views

Inverse of a matrix with the first standard basis vector as its first column

Consider an invertible matrix $n$ by $n$ matrix of the shape $$\begin{bmatrix}1&0\\c&A\end{bmatrix},$$ where $0$ denotes the $n-1$ zero column vector and $A$ is an $n-1$ by $n-1$ matrix. Is ...
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2answers
38 views

Determine equation of tangent plane?

Determine equation of tangent plane in points $(\frac{1}{2},1,f(\frac{1}{2},1) )$ $f(x,y)=x^{4}-x^{2}+y^{2}$ I know usually how these examples work, but I am confused with these $3$ points. I have ...
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Linear fit statistics

I'm a programmer, and in my code I use a linear fit function that currently just returns the resulting coefficients. I've now been asked to provide additional "statistics" relating to the fit, but I ...
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2answers
94 views

Surface described by the equation $-3y^2 - 4xy + 2xz + 4yz - 2x - 2z + 1 = 0$

Given the equation : $-3y^2 - 4xy + 2xz + 4yz - 2x - 2z + 1 = 0$. Check if the surface described by that equation has a center of symmetry and then by making the correct coordinate system change, find ...
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2answers
33 views

$T$-invariant subspaces, where the characteristic polynomial of $T$ is $x^4-3x^3$

Let $T:V\to V$ be a linear transformation of a four-dimensional real vector spaces $V$. Assume that the characteristic polynomial of $T$ is $x^4-3x^3$. Show that $V$ hash $T$-invariant subspaces of ...
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Linear maps (find transformation matrix)

For any polinomial $p \in \mathscr P^2$ let $$q_p(t) := p(-1)+p(0)t + p(1)t^2 + p(2)t^3.$$ Consider the $\varphi\colon \mathscr P^2 \to \mathscr P^3$ defined by $\varphi(p) := q_p.$ Prove that the ...
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1answer
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How can I calculate the angle of a line/vector if the center of the image is not (0,0)?

Simple image about the problem How can I calculate the alpha? My center of the image is (320,240) because it is a 640x480 image and the upper left corner is the (0,0). I tried to calculate it with ...
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2answers
40 views

Diagonalizable by orthonormal matrix

Given the matrix $$A = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$ Explain why $A$ can be diagonalized by an orthonormal matrix and find an ...