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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Rational solutions to a system of equations

I have a system of equations $$\begin{align} xy + 3zw = 0; \\ xz + 2yw = 0; \\ xw + yz = 0. \\ \end{align}$$ Plugging it into a CAS, I see that all the rational solutions to this system have ...
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1answer
181 views

Differentiating a Quadratic Form

I'm having some trouble differentiating a quadratic form. I'm tasked with showing that $P(x) = \frac{1}{2} \left(b-Ax\right)^T C (b-Ax)$ is minimized by a vector $x$ satisfying $A^T C A x = A^T C b$. ...
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2answers
1k views

Prove that if two vectors are parallel, one is a scalar multiple of the other

Im working with the following definition: Two vectors, $\vec{x}$ and $\vec{y} \in \Bbb R^n$ are parallel iff $|\vec{x} \cdot\vec{y}|=\|\vec{x}\|\|\vec{y}\|$ Then, I must prove that if two vectors ...
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0answers
19 views

Probablistic bound for $\|RR^TM\|$ for uniformly random orthonormal matrix $R$

I am stuck on a finding a probablistic bound on a nonstandard random matrix. I looked around on the internet and couldn't find any results. This could be because I don't know the key words or because ...
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1answer
40 views

$\varphi(f)$ is invertible iff $f$ is non-degenerate?

Let $E$ be the vectorial space of the bilinear functions $\varphi: \mathbb R^n\times \mathbb R^n\to \mathbb R$. Then, there is a canonical isomorphism between $E$ and the set of the real matrices ...
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236 views

Transformation from cartesian to polar Coordinates of Vector Field

This is fairly low-level, still I would like to get a verification: I vector field $$\mathbf{F}=F_x \hat{e_x} + F_y \hat{e_y} = F_r \hat{e_r} + F_{\phi} \hat{e_\phi}$$ given in cartesian coordinates, ...
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4answers
841 views

Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
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1answer
37 views

How do I prove (scalar1 + vector1) * scalar2 is not equal to scalar1 * scalar2 + scalar2 * vector1?

I am taking a Linear Algebra course and have been stumped on a homework question for a few hours. How do I prove for two scalars, $c_1$ and $c_2$, and a vector $v$: $(c_1 + v)c_2 \neq c_1c_2 + c_2v$ ...
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1answer
50 views

Sympletic matrix must have a determinant equal to one. [duplicate]

I hope that I am just confused, but I don't see why a sympletic map must have a determinant equal to one and not minus one?-Could anybody help me with that?- I am referring to the group $$Sp = \{T ...
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1answer
42 views

Injective endomorphism on a finite field is surjective?

Can you guys give me any hint on how to prove(or disprove): any injective endomorphism on a finite field is also surjective?
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2answers
79 views

Showing that a matrix is invertible and finding its inverse

I'm incredibly rusty at linear algebra, and in preparation for my course I've been doing some review questions. I've been staring at this one for a half hour and still don't know how to approach it: ...
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3answers
121 views

System of Linear Equations - how many solutions?

For which real values of t does the following system of linear equations: $$ \left\{ \begin{array}{c} tx_1 + x_2 + x_3 = 1 \\ x_1 + tx_2 + x_3 = 1 \\ x_1 + x_2 + tx_3 = 1 \end{array} \right. $$ ...
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2answers
55 views

Basis Represented By Standard Basis

Is it true that all basis vectors can be represented by the standard basis? for example every basis in $R^2$ like $(-2,1),(1,1)$ are linear combination of $(1,0),(0,1)$
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38 views

Which theorem can I use for prove if a matrix is invertible or not?

Let $v1,v2,...,v_n$ column vectors of invertible matrix $A \in \mathcal{M}_{n\times n}$ with coefficients in $\mathbb{R}$ and $M \in \mathcal{M}_{n\times n}$ such that the i-th column vector of $M$ ...
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2answers
180 views

Proving that if a system of linear equations has more than one solution, it has infinite solutions

I am a bit confused about the following proof, and have a question about the last step: First, assume there is more than one solution. Suppose $(c_1, c_2, ..., c_n), (d_1, d_2, ..., d_n)$ are two ...
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1answer
73 views

If $I + A + \cdots + A^{n-1} = O$, $A$ a square integer matrix, $n$ odd, for what $k$ does $\det(\sum_{i = k}^{n-1} A^i) = \pm 1$?

This question is, in a sense, homework. I'm taking a problem-solving seminar which uses questions like these, the first question on the 2010 Virginia Tech Regional Math Competition, as fodder. The ...
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2answers
105 views

Jordan form of a Matrix with Ones over a Finite Field

Question: Find the Jordan Form of $n\times n$ matrix whose elements are all one, over the field $\Bbb Z_p$. I have found out that this matrix has a characteristic polynomial $x^{(n-1)}(x-n)$ and ...
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1answer
293 views

Proof of Laplace expansion using minors

I've come across with the following proof of the Laplace expansion: Let $\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$ and $\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar ...
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1answer
51 views

Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
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32 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
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2answers
36 views

Curve in union of hyperplanes

If a smooth curve $\gamma: [0,T] \to \mathbb R^n$ is contained in the union of hyperplanes $$ \bigcup_{i=1, \dots, N} H_i$$ does it then follow that one can always find time intervals $[t_0, t_1]$ ...
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1answer
54 views

Algebraic Combinatorics about a Finite Graph

Here is a problem listed on a book 'Algebraic Combinatorics' by Richard P.Stanley. Let $G$ be a finite graph with at least two vertices. Suppose that for some $l \ge 1$, the number of walks of ...
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1answer
101 views

If $A^n = I$, $n$ odd, $A$ a square integer matrix, does $A = I$?

Edit: Crap, even my hypothesis was wrong. If you put $A = \left[ \begin{array}{cc} 1&-1\\3&-2 \end{array} \right]$, then $A^3 = I$ but no eigenvalue is $1$. (What's true is that all ...
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1answer
113 views

Orthonormal basis for orthogonal complement

I need to find the orthonormal basis for the orthogonal complement of $U = sp{(3,12,-1)}$ (U is a subset of $R^3$). This is what I have done so far: $V^⊥ = {(x,y,z)|(x,y,z)(3,12,-1)=0}$ ...
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1answer
44 views

Three equalities and one distinctness condition for four matrices

Are there matrices $A,B,C,D$ (in $M_n(K)$ for some $n$ and $K$) such that $$ AB=BA, \ CD=DC,\ AD-BC=I_n, \ DA-CB\neq I_n $$ It follows from what’s shown in that MSE question that $A,B,C,D$ cannot ...
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1answer
57 views

How prove this two condition is equivalent

Let $\alpha\neq \beta$ be nonzero column vectors in n-dimensional Euclidean space $\mathbb{R}^n$. Show that this follow two conditions are equivalent (1): $\alpha^T\beta>0$ (2): there exists ...
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1answer
57 views

Algorithm to determine if integer matrix is similar to symmetric integer matrix with nonnegative entries

Let $A\in M_n(\mathbb{C})$ be a matrix with integer entries (treated as a matrix over the complex numbers). Is there an efficient way to check if $A$ is similar to a symmetric matrix with nonnegative ...
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2answers
88 views

Matrix Algebraic Operations, If AA = AB, does A = B?

A and B are 2 x 2 matrices and A is not a zero matrix. How is the following proof incorrect? Since AA = AB, AA - AB = 0 A (A - B) = 0 and since A does not equal zero, then A - B = 0, therefore A = ...
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1answer
164 views

Tridiagonalize matrices with Householder transformation

I know that it is possible to tridiagonalize symmetric matrices by using a Householder trafo. I also found that we can get any matrix to Hessenberg form by using Householder trafos, but I still don't ...
3
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1answer
38 views

Partition of a Matrix

In Linear Algebra, we have been taught that the partition of a matrix $A$ consists of matrices,or blocks. In other words, its elements are matrices. This same, partitioned matrix, however is said to ...
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1answer
223 views

Can any two disjoint nonempty convex sets in a vector space be separated by a hyperplane?

Let $V$ be a normed vector space over $\mathbb{R}$, and let $A$ and $B$ be two disjoint nonempty convex subsets of $V$. A geometric form of Hahn-Banach Theorem states that $A$ and $B$ can be separated ...
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1answer
44 views

Linear algebra and substitution

Given matrix $$A =\begin{bmatrix} 3 && 1 \\ 2 && 1\end{bmatrix}$$ Compute $p(A)$, where $p(x) = x-2$. How do I go about doing this?
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1answer
169 views

Should I change my Linear Algebra Textbook?

I know there are many questions related to linear algebra, but the textbook I'm using is not that widely used as other books, I guess. My university uses 'Finite-Dimensional Linear Algebra' by Mark ...
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2answers
72 views

How does the max of $\prod_i a_i$ work?

Here are two succinct statements of the 'same' question: Statement 1: Take $a>0$ and $S \subseteq \mathbb{R}^N; S=\{(x_1,\dots,x_N)| \frac{1}{N}\sum_i x_i = a; x_i>0\}$. Define a 'product ...
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2answers
47 views

If $a+(1/(a-2))=4 $ then $(a-2)^2+(1/(a-2))^2$ is .

If $a+(1/(a-2))=4 $ then $(a-2)^2+(1/(a-2))^2$ is . Note: $a^2+(1/(a-2))^2=4^2$
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2answers
45 views

$\dim \mathscr L(\mathbb R^m,\mathbb R^n;\mathbb R^p)$

I know we can identify the set $\mathscr L(\mathbb R^m;\mathbb R^n)$ of the linear transformations $f:\mathbb R^m\to \mathbb R^n$ with a matrix $M_{n\times m}$ in the canonical way. Then, we have ...
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3answers
231 views

Derivative of spectral norm of symmetric matrix

I want to calculate the derivative of the spectral norm of a symmetric square matrix $W$: $$ \frac{\partial}{\partial w_{ij}} \|W\|_2 $$ How should I go about this?
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0answers
79 views

Question about condition number $k$ of a matrix over a finite field

If $\lambda_{max}$, and $\lambda_{min}$ denote the maximum and minimum values of the eigenvalues of a normal square matrix repectively- are there any explicit bounds to the eigenvalues of such a ...
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2answers
128 views

Determinant of a matrix with trigonometry functions.

Prove that the matrix is invertible for any value of $\beta$. I've done several exercises of this type. But I'm not sure with this one: $$\begin{bmatrix}\cos \beta & \sin \beta & 0\\ ...
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1answer
50 views

Finding a minimal polynomial of a linear operator

Given a linear operator, is there a straightforward way to find a minimal polynomial? I just learned that minimal polynomial can give information about the diagonalizability of the matrix, but I ...
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1answer
173 views

General solution for intersection of line and circle

If the equation for a circle is $|c-x|^2 = r^2$ and the equation for the line is $n \cdot x=d $, and assuming that the circle and line intersect in two points, how can I find these points? Also as ...
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0answers
226 views

Vector space basis change: is this “index-free” notation correct?

There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ...
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1answer
26 views

Identifying a solution in a linear system with generalized values

I am given three equations with two unknowns. I am asked to find an equation which must be satisfied by $a$, $b$, & $c$ in order for this system to have a solution. $$ x - y = a \\ -2x + 3y = ...
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1answer
117 views

When is there a vector $D$ with positive coordinates such that $e^{Ct}D$ has a negative coordinate?

Let $C$ be a $2 \times 2$ asymmetric matrix with real entries. Assume that $C$ has strictly negative, real eigenvalues. Fix $D\in\mathbb{R}^2$, where $D > 0$ (i.e., both coordinates are strictly ...
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1answer
370 views

Find parametric equations using parallel lines and line through a point

How would I find the parametric equation of a line through $(1,-1,1)$ and parallel to the line $x + 2 = 1/2y = z -3$. Would I find the vector equation first? If so, how would I go about doing that?
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1answer
37 views

Case Deletion Diagnostics

I have NO idea how to approach this problem. I don't see any connection between the corollary and the formula we need to prove. Does anyone have any hints? Corrolary: If $\mathbf{A}$ and ...
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1answer
137 views

Solving non-square linear systems with the exterior product and Cramer's rule

I'm reading the book Linear algebra via exterior products by Sergei Winitzki (which is the worst book, ever) and he shows that you can solve linear systems with a general solution with Cramer's rule ...
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2answers
134 views

How to prove $\exp(\ln M)=M$

Given a $n\times n$ real (complex) matrix $A$. Let me define: $$\exp A=\sum_{n=0}^\infty \frac{A^n}{n!}$$ and $$\ln A=\sum_{n=1}^\infty (-1)^{n+1}\frac{(A-I)^n}{n}$$ Let assume that the $2$ above ...
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1answer
59 views

$\langle Ax,x \rangle$ $\geq 0$, then $Ax=0$ iff $A^{t}x=0$

My friend asked me a question but I somehow stuck. Prove that if $A$ is a real $n*n$ matrix and $\langle Ax,x \rangle$ $\geq 0$ for all $x$ in $R^{n}$, then $Ax=0$ iff $A^{t}x=0$ I found that ...
2
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0answers
26 views

Find solution to matrix sandwich product [duplicate]

For any two $n \times n$ real symmetric and positive definite matrices $B$ and $C$, is it always possible to find a third real symmetric and positive definite matrix $A$ such that $ABA=C$? If not, ...