Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a ...
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Finding the 2 point coordinates for a known edge.

Say I have an edge A'B' which is a vector (5,3 9). How can I find the individial points A' and B' from A'B'. I translated the points A and B by a vector then combined them to make the edge AB. Then ...
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Problems on vector spaces

Let $E$ a $\mathbb{K}$-vector space of finite dimension $n$, $\mathcal{V}$ a subspace of $\mathcal{L}(E)$ such that $$\forall u\in\mathcal{V}\setminus \{0\},u\in\mathcal{GL}(E)$$ a) Show that ...
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Same quadratic forms on $\mathbb R^n$

Let $q$ be an inner product on $\mathbb R^n$ and $Q$ be its matrix expressed in the canonical basis of $\mathbb R^n$. Assume that the group $$SO(q)=\{A\in M_n(\mathbb R) \ | \ A^TQA=Q\}$$ of ...
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Show linear independence

Is the Set $$S=\{e^{2x},e^{3x}\}$$ linearly independent?? And answer says Linearly independent over any interval $(a,b)$,only when $0$ doesnot belong to $(a,b)$ How do I proceed?? Thanks for the ...
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A determinant problem

If $f(n)=\alpha^n+\beta^n$ and $$A=\left| \begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array} \right|$$ ...
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I am not getting the concept actually of dimension of intersection of subspaces

Let $$W_1=\{(0,x_2,x_3,x_4,x_5)|\forall x_i\in \Bbb R, i=2,3,4,5 \} $$ $$W_2=\{(x_1,0,x_3,x_4,x_5)|\forall x_i \in \Bbb R, i=1,3,4,5\}$$ be subspaces of $\mathbb{R}^5$ then what is $\dim(W_1 \cap ...
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Characteristic polynomial factor over the real numbers

Ve=the set of symmetric 2x2 matrices I'm trying to show that any element of Ve has a characteristic polynomial that factors over the real numbers and has two distinct eigenvalues unless the matrix ...
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1answer
15 views

Unique linear combination and basis

Let $S \in \mathbb{R}^3$ be the following set of vectors $$ v_1=\begin{pmatrix} 1\\ 0\\ 1 \end{pmatrix} , v_2 =\begin{pmatrix} 0\\ 1\\ 1 \end{pmatrix}, v_3 = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix} ...
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Is the largest eigenvalue a unique weighted sum of the linear combination of the elements of a matrix?

Let $\lambda$ be the largest eigenvalue of $\boldsymbol{A}\in\mathbb{C}^{n\times n}$ ($\boldsymbol{A}$ is hermitian). Is $$\lambda = ...
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Inverse of a diagonal matrix plus a constant

I am looking for an efficient solution for inverting a matrix in the following form: $D+aP$ where D is a (full-rank) diagonal matrix, a is a constant, and P is a one matrix. This question Inverse of ...
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Explicit Isomorphism between Vector Spaces

Let $V$ and $W$ be two finite dimensional spaces. I want to show that I have a canonical isomorphism from the space of bilinear forms $\mathcal{B}= \left\lbrace B: V^* \times W^* \rightarrow ...
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dimension of a symmetric matrix with trace zero

what will be the dimension of symmetric matrix of order $n\times n(n\geq2)$ with real entries and trace is equal to zero? The answer is given as :$\frac{n^{2}+n}{2}-1$ can anyone explain how will ...
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Is Im(T) + Ker(T) the same as Im(T) union Ker(T)

If i know Im(T) and Ker(T), is Im(T) + Ker(T) the union of the two vector space? If not, how do i find the addition of the two vector space. It is best if examples can be given. Thanks.
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Why is $\dim(W)=3$?

I am teaching myself upper-division linear algebra for the moment, and I currently do not understand this example in my textbook. This is from page 50 of Linear Algebra by Friedberg, Insel, and ...
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Number of possible 3x3 matrices with 0,1 entries [on hold]

I have a $3\times 3$ matrix whose entries can be $0$ or $1$. How many patterns can I make with this? I know it has something to do with the binomial coefficient, but I haven't studied it in Yeats.
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$|y-x|\lt \epsilon$ and $|z-x|\lt \epsilon$

I'm trying to solve this question: Is it not obvious we can arbitrarily approximate the points $y$ and $z$ to $x$? but how to formalize this? Any help? hints to begin with? Thanks Notation ...
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1answer
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Jacobian and PDE

I am wondering how to compute the Jacobian in order to know if a given PDE satisfying an initial condition has a unique solution or not. If I consider the PDE, $u_x=1$, satisfying the initial ...
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solve the homogeneous system (A-I)x=0 and use the result to solve Ax=x for x=[x1, x2, x3]

5. solve the homogeneous system $(A-I)x=0$ and use the result to solve $Ax=x$ for $x=[x1, x2, x3]$ Im pretty damn confused about what im supposed to do here, I computed $A-I$ and multiplied it by ...
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Proving idempotent

Let A be an involution and let B=1/2(I+A), C=1/2(I-A). Show that B and C are both idempotent and BC=0. TO SEE WHAT I HAVE TRY Visit: ...
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Show that if $A^{k+1}=0$ for some nonnegative integer $k$ then $A$ is singular…

Let $A$ be a $n \times n$ matrix. Show that if $A^{k+1}=0$ for some nonnegative integer $k$, then $A$ is singular, but $I−A$ is nonsingular and $(I − A)^{−1} = I + A + A^2 + \cdots + A^k$.
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Any segment can't be inside the sphere

I'm trying to prove if two points $a$ and $b$ are in the closed ball, then the segment between them is inside the ball, and can't be in the sphere, in another words: Let $a,b\in \mathbb R^n$, $a\neq ...
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show that for the system to be consistent we must have b2 = b3 - 2b1 [on hold]

The first problem, #4. I dont know how to approach this problem, I can see how the statement b2 = b3 - 2b1 is true but how do I prove it must be true?
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Can a matrix span $\mathbb{R}^3$

Can a matrix with 4 columns and 3 rows span $\mathbb{R}^3$? Does the following matrix span $\mathbb{R}^3$? $$\begin{bmatrix} 2 & 1 & -3 & 5\\ 1 & 4 & 2 & 6\\ 0 ...
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Decomposing into symmetric and skew-symmetric matrices

Let $V_e$ be the set of symmetric $n \times n$ matrices, let $V_o$ be the set of skew-symmetric $n \times n$ matrices, then $V_o$ and $V_e$ are a subspace of $V$, the space of $n \times n$ matrices.. ...
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1answer
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isomorphic mapping on direct sum and products

I found an old post here: https://www.physicsforums.com/threads/question-about-isomorphic-mapping-on-direct-sums.709423/ While reading the answer to the question posted in the link above, I found ...
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1answer
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$\langle Tx,Ty\rangle=\langle x,y\rangle$ for every $x,y \in H$. implies $T^*=T^{-1}$

Let $T:H \to H$ be a linear operator. $T$ is called isometry if $\|T(x)\|=\|x\|$. Prove that the following are equivalent: T is an isometry $\langle Tx,Ty\rangle=\langle x,y\rangle$ for every $x,y ...
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Symmetric and Skew Symmetric Matrices

$V_0=$ the set of $2\times2$ skew symmetric matrices I know that any element of $V_0$ has a characteristic polynomial that will not factor over the real numbers, and therefore has no eigenvectors. ...
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1answer
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Find the Basis and Dimension of a Solution Space for homogeneous systems

I have the following system of equations: x+2y-2z+2s-t=0 x+2y-z+3s-2t=0 2x+4y-7z+s+t=0 Which forms the following matrix ...
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Confusion over subspace dimension

I have a subspace $W$ of $\mathbb{R}^5$ where $W$ is the set of all $5$-tuples that satisfy the following conditions: ...
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Question about determinant as transformation on alternating multilinear n-forms

If $T:E\rightarrow F$ is a linear transformation, $f:F\times\cdots\times F \rightarrow \mathbb{R}$ is an alternating, multilinear n-form and $\tilde{T}:A_n(F)\rightarrow A_n(E)$ is a function that ...
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Consistent linear system of 3 equations and 4 unknowns [on hold]

How many solutions does a consistent linear system of 3 equations and 4 unknowns have? And why? (Plz respond ASP)
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For which values of the parameter $ a \in \mathbb{R}$ is the transformation $T(x,y,z)=(x+ay,x-ay,x+z)$ an isometry?

Find the eigenvalues and eigenvectors of the linear transformation $T(x,y,z)=(x+y,x-y,x+z)$. Verify that the eigenvectors are orthogonal. $$T(x,y,z)=\begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 ...
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1answer
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Inverse matrix proof

Let $A$ be a $n \times n$ matrix. Show that if $A^2=O$ then $A$ is singular, but $I−A$ is nonsingular and $(I−A)^{-1}=I+A$. What I have tiedy: $(I-A)*(I+A)=I-A+A-A^2$ $=I-A^2$ $=I-0$ since A^2=0 ...
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1answer
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f is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^−1 (V ))$ = V prove?

$f$ is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^{−1} (V ))$ = $V$ This is an assertion and i said it was true. But i am confused as to what is referred to as the domain and range in this question. I would say ...
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Solution to homogeneous linear differential equation form a vector space

Show that the solutions of a homogeneous linear differential equation $y"+a(x)y'+b(x)y = 0$ form a vector space. What is its dimension? I understand that the dimension is 2 and that 0 is a solution ...
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Find the inverse with respect to the binary operation $a ∗ b = a + b + a^2 b^2$

A binary operation on $\mathbb{R}$: $a * b = a + b + a^2 b^2$ The neutral element I found to be $0$. Then I need to find an invertible element having two distinct inverses. I don't know where to ...
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Skew Symmetric Matrix Properties

We have a theorem says that "ODD-SIZED SKEW-SYMMETRIC MATRICES ARE SINGULAR" . Proof link is given here if needed. Now let us assume we have a $3\times 3$ skew symmetric matrices of the form $ ...
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Product symmetry matrix proof

Let A and B be symmetric n×n matrices.Prove that AB = BA if and only if AB is also symmetric. So we need to prove that AB is symmetric. This means (AB)^T=AB. Recall a property of transposes: the ...
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Issues with connecting the SVD and Eigenvalues for block matrix

In class, we have talked about the singular value decomposition and its connection to Eigenvalues. Specifically, for a matrix A, if the columns of a matrix contain linearly independent eigenvectors, ...
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Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
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Matrix Representation of a linear operator and Checking if it is surjective and injective

Let $W_1$ be the subspace of C(0,1) spanned by the functions $\{e^x,xe^x,x^2e^x\}$. Let $W_2$ be the subspace of C(0,1) spanned by the functions $\{1,e^x,xe^x,x^2e^x\}$. Let T be the application ...
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Diagonalizing $xyz$

The quadratic form $g(x,y) = xy$ can be diagonalized by the change of variables $x = (u + v)$ and $y = (u - v)$ . However, it seems unlikely that the cubic form $f(x,y,z) = xyz$, can be ...
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matrix singular proof

Let A, B be n×n matrices. Show that if AB = A and B≠I then A must be singular. I was thikning to prove it by contradiction, showing if A is nonsingular then we have thta AB=BA=A, therefore B is the ...
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How to proof the cofactors expansion for determinants? [duplicate]

I have a homework, where I must prove that the cofactors expansion for determinant is equal to the permutations definition of it. I have been thinking about it and my first idea is by induction in the ...
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How to efficiently solve a series of similar matrix equations using the LU decomposition

This is the problem I'm dealing with: Let $\sigma_1,\dots,\sigma_n \in \mathbb{R}$ and $b_1,\dots,b_n$ be column vectors of length $n$. Consider the system $$ (A - \sigma_jI)x_j = b_j, \quad ...
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Subsets that are also vector spaces

The vector space $R^3$ and the subset M consists of the vectors $(\xi_1,\xi_2,\xi_3)$ for which i) $\xi_1 = 0 $ ii) $\xi_1 = 0$ or $\xi_2 = 0 $ iii) $\xi_1 + \xi_2 = 0 $ iv) $\xi_1 + \xi_2 = 1 $ ...
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Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
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Is my answer correct?

I'm trying to solve this question: My solution: Since $\varphi$ is continuous we have: $C\text{ is convex}\implies C\text{ is connected}\implies \varphi(C)\text{ is connected}\implies \varphi(C) ...