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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How to solve $\frac12 \sec^2 \frac x2 = 1$ under restricted domain?

solve: $$\frac12 \sec^2 \left(\frac x2\right) = 1$$ and domain $x: (-\pi,\pi) \cup (\pi,3\pi)$. sec^2 (x/2) = 2 sec^2 (x/2) can be re-written as tan(x/2)^2 + 1, therefore tan^2(x/2) + 1 = 2 ...
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3answers
346 views

Find a matrix $P$ that orthogonally diagonalizes $A$ and determine $P^{-1}AP$

I'm completely lost on how to finish this problem. The problem from the book is Find a matrix $P$ that orthogonally diagonalizes $A$ and determine $P^{-1}AP$ and my matrix is the $2 \times 2$ ...
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1answer
64 views

Error with thinking about shortest distance between two lines in three space

Given the vector equations for two lines $$(x_0,y_0,z_0) = (a,b,c)+t(d,e,f)$$ and $$(x_1,y_1,z_1) = (g, h, i) + t(j, k, l),$$ why is the shortest distance between the two lines not equal to ...
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1answer
57 views

Confusion over Matrix rotation

I want to make a function in C++ that accepts an angle 'a', and a vector 'v' as arguments and returns a matrix. 'a' should represent the amount that is rotated around vector 'v', an arbitrary axis, ...
2
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2answers
237 views

orthogonal projection onto orthogonal complement

If $V=M \oplus M^{\perp}$. For any $v\in V$, the orthogonal projection of $v$ onto $M$ along $M^{\perp}$ is well defined. Can we take the orthogonal projection of $v$ onto $M^{\perp}$ along $M$?
2
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1answer
124 views

Linear Algebra 101 - Optimizing inequalities

I am considering the region contained in $\mathbb{R}^2$ consisting of all the points that satisfy all the following inequality: $-4 \leq y < 4 \\ -9 \leq 2x + y \leq 9 \\ -9 \leq x + 2y \leq 9 \\ ...
2
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1answer
89 views

How to use Sylvester's law of inertia to show positive definiteness

Let $G = (<e_i,e_j>)_{i,j=1,...3} :=\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{matrix} \right)$ be a symmetric bilinear form $< \cdot, ...
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2answers
53 views

Prove that the linear space of polynomials with root $\alpha \in \mathbb{R}$ is a subspace of $\mathbb{R}[x]_n$

Prove that linear space of polynomials having root $\alpha \in \mathbb{R}$ is a subspace of $\mathbb{R}[x]_n$. It's also required to find basis and dim of that subspace. I recently started learning ...
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1answer
43 views

Subgroups of GL(2,C) isomorphic to Z

Let $\mathbb Z\to \mathrm{GL}_2(\mathbb C)$ be an injective homomorphism. I'm wondering about the possibilities for the image of $\mathbb Z$. I think the image is always conjugate to a subgroup of ...
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1answer
34 views

Proof regarding effect of row operations on determinants>

Let $A,B \in K^{n,n}$ and suppose $B$ is obtained from $A$ by adding $\lambda$ times row $j$ to row $i$. Prove $det(A)=det(B)$. My Attempt I tried to use proof by induction for this . Take ...
2
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4answers
281 views

Eigenvalues for $AB$ and $BA$ when $A$ and $B$ are not square

Let $A$ and $B$ be $m\times n$ and $n\times m$ complex matrices, respectively, with $m < n$. If the eigenvalues of $AB$ are $\lambda_1, \ldots, \lambda_m$, what are the eigenvalues of $BA$? If ...
0
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1answer
74 views

$A \cdot B = A \cdot C$ does not imply that $B = C$

I am trying to prove the following: If $A,B,C$ are non-zero vectors such that $A \cdot B = A \cdot C$, then it's not necessarily true that $B = C$. My proposed proof. Suppose $A \cdot B = ...
0
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1answer
42 views

Eigenvalue decomposition of $2\times 2$ matrix

There is a matrix $$A = \begin{bmatrix}3 & 1\\ 1 & 3 \end{bmatrix}$$ and the factorisation of $A = S*D*S^T$ needs to figured out. So far I have figured out that the eigensolutions are 4 and 2 ...
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1answer
40 views

Augmented form of three equation with one extra unknown

I am being asked to find all solutions to the system of linear equations: $$ 2u+v−w+z = 1\\ v−2w = 2\\ w+z = −1 $$ The problem is that there is an extra term $2u$. Ignoring $2u$ the augmented matrix: ...
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3answers
124 views

Find eigenvector of the linear operator

Task is to find an eigenvector of the following linear operator: $f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$. I know how to find eigenvectors ...
0
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1answer
37 views

Calculate a Basis of the subspaces U and V

$V=\{\big( 1,0,2 \big),\big( -2,0,5 \big)\} $ $U=\{\big(-1,0,3 \big)\} $ Calculate a basis of the subspaces $U+V$. My approach is to show that the vectors are linearly independent and then I have my ...
2
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1answer
79 views

Can some one explain me a easy alternate proof of rank nullity theorem?

OK I have just gone through the Gilbert Strang's 'Introduction to Linear algebra'. I felt it as a very nice book. But I am bad at proving things. I have an idea what null space (kernel) and column ...
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1answer
37 views

Minimum of general quadratic forms

I am asking to find the minimum value of the following quadratic form $$ \left|\left|x-Mz\right|\right|_2^2=\langle x-Mz,x-Mz\rangle=z^TM^TMz-2(M^Tx)z+x^Tx $$ where $x$ is a constant vector, $M$ is ...
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1answer
27 views

Non degenerate upper triangular matricies 2

Consider the set V of upper-triangular n×n matrices with elements in some field K. I.e., if A is such a matrix, aij=0, for i>j Show that non-degenerate upper-triangular matrices form a group with ...
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3answers
31 views

Upper Triangular Matricies

Find the determinant of upper triangular matrices (where $a_{ij}=0$ when $i>j$) I am not sure how to calculate this using the definition of a determinant. It is the bit with the permutations ...
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0answers
112 views

Meaning of $A^A$

Let $A$ be a matrix. I know that there is definition for $A^k$ power of $A$, and $e^A$ expontential of $A$. Is there any meaning of $A^A$?
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2answers
96 views

Minimal polynomials for a matrice in a given field and in its extension [duplicate]

For given matrices $X$ from the matrix ring $M_{n,n} (F)$, where $F$ is a field, the minimal polynomial for $X$ in $F[x]$ is the polynomial $P$ of the lowest degree with the following properties: ...
4
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2answers
119 views

Non-integral power of a singular matrix

I know, that if $A$ is nonsingular matrix, so $\det{A} \ne 0$, then $A^p=\exp\left(p\ln A\right)$ is true for any real exponent, but what about if $A$ is singular? Then $A$ has a zero eigenvalue, so ...
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0answers
44 views

Entries of tensor product

If I have a pure tensor $v_1\otimes\dots\otimes v_n$ in an $n$-fold tensor product $V^{\otimes n}=V\otimes\dots\otimes V$ what is the canonical name for the vector $v_j$? Do I call it the $j$-th entry ...
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1answer
24 views

Show that the system of pairs of basis vectors is not a basis for the cartesian product

Let $V, W$ be vector spaces over an arbitrary field $K$, $B= (b_i)_{i \in I}$ basis of $V$, $C=(c_j)_{j \in J}$ basis of $W$. I want to show that the system $((b_i,c_j))_{(i,j) \in I \times J}$ ...
1
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1answer
51 views

How to guess the basis

This may be a silly question, but still I'm asking you. Suppose you are given a vector space. How do you guess the basis? I think is this not always easy to guess the maximal linearly independent set ...
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0answers
40 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., $\begin{bmatrix}0 ...
0
votes
1answer
30 views

Determinant reduction problem

If a,b,c are all different and $\displaystyle \begin{vmatrix} a &a^3 &a^4-1 \\ b &b^3 &b^4-1 \\ c &c^3 &c^4-1 \end{vmatrix}=0$, then show that abc(bc+ca+ab)=a+b+c. I try ...
2
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0answers
23 views

Rotations in a multidimensional space using jordan canonical forms

I asked a teacher if he could tell me some usefulness of a jordan canonical form, and his answer was: "In a multidimensional space, you could use a JCF to define rotations for each of the planes." ...
4
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1answer
79 views

vectors with the same mutual angles

Let $S = \{v_1,\ldots,v_n\} \subset \mathbb{R}^n$ and let $T = \{w_1,\ldots,w_n\} \subset \mathbb{R}^n$ be such that the angle between $v_i$ and $v_j$ is equal to the angles between $w_i$ and $w_j$. ...
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0answers
32 views

Can Anyone Check my Proof that all linear maps $T:U_1\oplus U_2\longrightarrow V_1\oplus V_2$ are of the form below?

Suppose $U_1, U_2$, $V_1$ and $V_2$ are vector spaces such that $$\textrm{dim}(U_1)=\textrm{dim}(V_1)=n-k\quad \textrm{and}\quad \textrm{dim}(U_2)=\textrm{dim}(V_2)=k$$ and ...
0
votes
1answer
28 views

The slope of a sum of linear equations

Assuming I have a set of linear equations: $$ y_1 = a_1x_1 + b_1 $$ $$ y_2 = a_2x_2 + b_2 $$ Is it correct that the sum of these are: $$ z1 = y1 + y2 $$ This then means that the y-intersect and ...
2
votes
1answer
109 views

Swapping rows or columns of Toeplitz matrix changes sign of one eigenvalue

Given some arbitrary Toeplitz matrix, if I swap two rows, one of the eigenvalues change its sign. For example, $X = \begin{bmatrix} A & B & C \\ D & A & B \\ E & D & A ...
2
votes
2answers
543 views

Eigenvalues in terms of trace and determinant for matrices larger than 2 X 2

The eigenvalues of a $2\times2$ matrix can be expressed in terms of the trace and determinant. $\lambda_\pm = \frac{1}{2}\left(\textrm{tr} \pm \sqrt{\textrm{tr}^2-4\det}\right)$ Is there a similar ...
2
votes
1answer
96 views

Proof that eigenvector corresponding to simple eigenvalue is continuous

Let $\lambda$ be a simple eigenvalue of $A \in L(C^n)$ and let $x$ be the corresponding eigenvector. Then for $E \in L(C^n)$, $A+E$ has an eigenvalue $\lambda(E)$ and an eigenvector $x(E)$ such that ...
3
votes
1answer
51 views

Convex hull of $\exp\bigl( \mathcal{M}_n(\mathbb R)\bigr)$

What is the convex hull of $\exp\bigl( \mathcal{M}_n(\mathbb R)\bigr)$ ? My attempt : lemma For $A\in\mathcal{M}_n(\mathbb C)$ there exist $P(X)\in \mathbb{C}[X]$ such that $A=\exp{P(A)}$ ...
1
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1answer
49 views

linear transformations with matrices $A, A^*$

Let $K$ be a field, $K\subseteq \Bbb C$. $V$ is a linear space over $K$, $\dim(V)=n(n\geq2)$. Choose ordered basis $\epsilon_1,\epsilon_2,\dotsc,\epsilon_n$ for $V$. $\bf A,B$ are two linear ...
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2answers
188 views

Derive a transformation matrix that mirrors the image over a line passing through the origin with angle $\phi$ to the $x$-axis.

Question: Using homogeneous coordinates, derive a $3$x$3$ transformation matrix $M$ that mirrors an image over a line passing through the origin, with angle $\phi$ to the $x$-axis. Comment: This is ...
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2answers
206 views

Reflection in a plane.

What is the exact definition of a reflection through the plane $a.r=0$ for a given vector a and $r=(x,y,z)$. Of course I know what it is but I don't know what's part of its definition and what's part ...
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2answers
75 views

The equation of the plane in five-dimensional space

Given a five-dimensional space. There are three points (coordinates) and need to find the equation of a plane through 3 points. How to do this? $$B(1,1,0,1,1)$$ $$C(8,7,3,1,4)$$ $$D(1,0,-1,3,-3)$$
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1answer
34 views

If vectors $m + s + l = 0$, then $m \times s = s \times l$

If vectors $m + s + l = 0$, then $m \times s = s \times l $ True or false, if true show it if false show an example. I assume true: So I have let vectors $u\|v$, defined by $u = cv$ The angle ...
0
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1answer
53 views

Ideal of an integral domain all of whose exterior powers are nonzero.

I want to find an integral domain $R$ with ideal $I$ (considered as an $R$-module) such that $\bigwedge^k I\neq 0$ for all nonnegative integers $k$. Dummit and Foote gave the example of $R=\mathbb ...
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3answers
66 views

$\det A \neq 0$. Prove that $\det A^* \neq 0$.

$A$ is matrix representing operator $\mathcal{A}$. $*$ is such operator that respects following equality: $(\mathcal{A}x,y)=(x, \mathcal{A}^*y)$; (I don't know what term is used in English). ...
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2answers
40 views

Linear Algebra determinant reduction 2

If $\begin{vmatrix} x &x^2 &1+x^3 \\ y &y^2 &1+y^3 \\ z &z^2 &1+z^3 \end{vmatrix}=0$, then prove, without expansion, that $xyz=-1$, where $x,y,z$ are not equal.
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2answers
53 views

Eigenvalues of Matrix with 1s everywhere but diagonal [duplicate]

I'm not sure if this type of matrix has a name, but I feel as if there's a trick to finding the eigenvalues that i'm missing: $$ a \in R $$ $$ M = \begin{bmatrix} 1 + a & 1 & 1 ...
0
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1answer
30 views

Linear Algebra Infinite dimensional bases

Let $V$ be a vector space over a field $K$. Suppose $V$ is the set of all infinite sequences $a_0$$a_1$$a_2$..., such that in each sequence only finitely many of the components $a_i$ $\in K$; i = ...
5
votes
5answers
254 views

Linear Algebra determinant reduction

Prove, without expanding, that \begin{vmatrix} 1 &a &a^2-bc \\ 1 &b &b^2-ca \\ 1 &c &c^2-ab \end{vmatrix} vanishes. Any hints ?
1
vote
1answer
44 views

Comparing two linear functions

Let $X$ be a Banach space and let $h:X\to\Bbb C$ and $f:X\to\Bbb C$ be two bounded linear functions such that if for some $x\in X$ we have $f(x)=0$ then $h(x)=0$. Prove that there exists a ...
3
votes
0answers
36 views

Problem about real square matrix with rank 1 [duplicate]

Given $A \in \mathbb{R}^{n \times n}$ and $\text{rank}(A) = 1$. By working only on real field, show that $A$ is diagonalizable if and only if $\text{tr}(A) \neq 0$. Here, $\text{tr}(A)$ is the sum of ...
2
votes
1answer
31 views

Prove that $\mathcal{AB}$ is linear operator if $\mathcal{A}$ and $\mathcal{B}$ are linear operators.

It is fairly easy to determine whether $\mathcal{AB}$ is linear when we know $\mathcal{A}$ and $\mathcal{B}$ (for example, $\mathcal{Ax}=(2x_1, 3x_2-x_1)$ and $\mathcal{B}$ is something similar). But ...