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 ...

learn more… | top users | synonyms (1)

0
votes
2answers
47 views

How to compute this determinant, without the Sylvester determinant theorem, [duplicate]

The problem statement is: Show that there exists numbers $a$ and $b$ such that $$det (A + sxy^*)= a+bs$$ here $A$ is an $nxn$ matrix with real entries, and $x,y\in R^n$. I've been using brute ...
0
votes
0answers
24 views

Find a direct basis for boundary orientation

Let M in R4 be a manifold defined by equation d= $x^2$ + $y^2$ +$z^2$ and oriented by sgn $dx_1$^$dx_2$^$dx_3$. Consider the subset where d<=1. Show that it is a piece with boundary. Let x be a ...
1
vote
1answer
38 views

Prove that dim$($Ker$(A^n))$ is odd for all $n$ sufficiently large

I have the following problem: Let $A$ be a matrix of $101$ by $101$ over $\mathbb{F}_2$, such that trace$(A^n)=0$ for all $n$ positive integer. Prove that dim$($Ker$(A^n))$ is odd for all $n$ ...
2
votes
2answers
40 views

Proof about Diagonalization of A

The question asks WHY is it true that $$A^{n} = PD^{n}P^{-1}$$ I can never do proper proving in algebra; what I almost know for sure is that a proof by induction is the way to go here. But how do you ...
7
votes
1answer
44 views

Prove that p divides to algebraic multiplicity of the eigenvalue

I need help in the following exercise of a qualifying exam: Let $A$ be a matrix of size $m$ by $m$ over the finite field $\mathbb{F}_p$ such that $\operatorname{trace}\left(A^n\right)=0$ for all $n$. ...
0
votes
1answer
41 views

QR Factorization for Inconsistent Linear System

I am trying to recreate the problem found here on finding the least squares solution to an inconsistent linear system via QR factorization. Can someone explain the part about adding on vectors so that ...
0
votes
1answer
31 views

make two sets of eigenvectors orthogonal

Suppose that $v_1,v_2,v_3$ are orthogonal eigenvectors of $A$ and $u_1,u_2$ are orthogonal eigenvectors of $B$. Is there any way to choose $v_1,v_2,v_3$ and $u_1,u_2$ such that $v_1,v_2,v_3,u_1,u_2$ ...
1
vote
1answer
43 views

Prove rayleigh quotient = operator norm without referring to eigenvalues

Let $H$ be a Hilbert and $T \in \mathcal{L}(H,H)$ a symmetric operator. Prove $$ \|T\| = \sup\{|(x,Tx)| : x \in H, \|x\| = 1\} $$ without referring to the eigenvalues of $T$ (which is what all ...
0
votes
1answer
29 views

Projection of Range of Matrix $ A $ onto Kernel of $ A-I$

Let A be a $2 \times 2$ matrix such that $A^2 = A$. Show that $Ax = x$ for every $x$ in $R(A)$ and if $rank(A)$ $= 1$, $M = R(A)$, $N = Ker(A-I)$ then $A $ is the projection along $N$ onto $M$. For ...
3
votes
0answers
136 views

Measure theory , Functional calculus, Self Adoint

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. I need to show $\langle g_a|g_b \rangle=0$ if ...
3
votes
0answers
34 views

There exists an algebraic basis $(e_i)_{i \in I}$ in $E$ such that $\|e_i\| = 1$ for all $i \in I$?

Let $E$ be a infinite-dimensional normed vector space. How do I see that there exists an algebraic basis $(e_i)_{i \in I}$ in $E$ such that $\|e_i\| = 1$ for all $i \in I$?
7
votes
3answers
688 views

How can Hamilton's quaternion equation be true?

I'm reading Ken Shoemake's explanation of quaternions in David Eberly's book Game Physics. In it, he describes the $\mathbf{i}, \mathbf{j}, \mathbf{k}$ components of quaternions to all equal $\sqrt{-...
0
votes
1answer
69 views

Show that $\operatorname{rank} A = 3$ [closed]

Matrix $A \in R^{3, 2015}$ is given. It is known that matrix $AA^{T}$ is invertible. Show that $\operatorname{rank} A = 3$. How to start this? What does the info that $AA^{T}$ is invertible gives us?...
1
vote
1answer
18 views

Altering solution to differential equation using exponential constant (linear algebra)

We have a solution for a differential equation which is of the form: $y(t) \LARGE = \frac{1}{C_1e^t+1}$ Since all the constant does is shift the y(t) to left or right, we can substitute $\large ce^{...
2
votes
1answer
46 views

How do you solve this recurrence relation/use it in a sequence to find it's GIF value?

The sequence {$x_k$} is defined by $x_{k+1} = x_k^2 + x_k$ and $x_1=\frac{1}{2}$. Now, if [.] denotes the greatest integer function, then which of the following options is correct: A) $[\frac{1}{x_1 +...
0
votes
1answer
56 views

Basis of image and kernel of matrix [closed]

Vectors $\overrightarrow{x}, \overrightarrow{y}, \overrightarrow{z} \in \mathbb{R}^{2015}$ are linearly independent. Find basis of image and kernel of matrix: $ A = \left [\overrightarrow{x}+\...
8
votes
2answers
104 views

Determinant of Tridiagonal matrix

I'm a bit confused with this determinant. We have the determinant $$\Delta_n=\left\vert\begin{matrix} 5&3&0&\cdots&\cdots&0\\ 2&5&3&\ddots& &\vdots\\ 0&2&...
0
votes
1answer
34 views

Multiple root of characteristic polynomial

Suppose that V is a linear operator on a finite dimensional vector space. If V and U are two independent eigenvectors with the same eigenvalue \lambda.Must Lambda be a multiple root of f(x),...
1
vote
1answer
42 views

Second degree parametric inequalities

I am asked to solve the parametric inequality and to find for which values of a every x is a solution. $$ (a+5)x^2 - 2x(a+1) + 2a - 4 \ge 0 $$ So in order every x to be a solution to the ...
1
vote
1answer
33 views

What are the valors of $4x_1-x_1^2+x_3^2$?

$x_1$ and $x_2$ $x_3$ real numbers such that $x_1<x_2<x_3$ are solutions of the equation : $x^3-3x^2+(a+2)x-a=0$ where a is real What are the valors of $4x_1-x_1^2+x_3^2$ After ...
2
votes
2answers
32 views

Lemma: $x \cdot (Ay) = (Ax) \cdot y$

As simple as this may sound, I just do not understand what this statement implies. An $n \times n$ matrix A is symmetric if and only if: $$\bar{x}.(A\bar{y}) = (A\bar{x}).\bar{y}$$ Why is this true,...
0
votes
0answers
84 views

Relation between annihilator of kernel and range of the transpose

Let $A:X\to Y$ be a continuous linear map. Let $A^*:Y^*\to X^*$ denote the transpose of $A$ (also referred to as the dual transformation of $A$) Let $\ker A ^⊥$ denote the annihilator of $\...
1
vote
1answer
23 views

Can solution spaces of linear systems of equations be considered vector spaces?

I have recently started learning linear algebra. I have come across the concepts of vector spaces and their bases. One type of problem I am now encountering is finding the basis and dimension of ...
2
votes
1answer
25 views

Prove that $\beta⟘U$ for some $\beta \in W/\{0\}$.

In an $n$-dimensional vector space $V$, there are given $m$-dimensional subspaces $U$ and $W$ so that $\alpha⟘W$ ($\alpha$ is orthogonal to every vector in $W$) for some $\alpha \in U/\{0\}$ . Prove ...
0
votes
2answers
23 views

If a is real , then what is the solution set of the inequality $ | x | \leq { a\over x } $?

Is there a shorter way than solving it in different cases : ie. $ x\neq 0 $ Case 1 : a > 0 Subcase 1 : x > 0 Subcase 2 : $ x < 0 $ Case 2 : a < 0 Subcase 1 : x > 0 Subcase 2 : $ x &...
0
votes
4answers
35 views

How do you distribute this negative?

So I have $-(x - 2)^2$. Do I rewrite it as $-(x - 2)\cdot-(x - 2)$ and distribute the negative to the inside making it $(-x + 2)(-x + 2)$ or add the negative at the end of doing FOIL?
2
votes
1answer
29 views

Condition on Lyapunov Equation output being positive definite

I have Matlab code that solves the Lyapunov equation $AX + XA^T + Q = 0$ for a 3-D array of matrices, $A$, using the Matlab function lyap(A,Q). My problem is that ...
1
vote
1answer
81 views

why S in SVD is a vector instead of a matrix?

I know that when applying SVD on a matrix (m * n) I should have these three outputs: ...
2
votes
5answers
94 views

If $A$ is a $3\times 2$ matrix and $B$ is a $2\times 3$ matrix why is $AB$ not invertible

I am getting nowhere with this question. I found a similar question on this site but I need help in a more step by step way. I would be very grateful.
0
votes
2answers
57 views

Given two polynomials $p_1$ and $p_2$, how to define polynomial $p_3$ that has all roots of $p_1$ except those that are $p_2$'s roots?

Hypothesis: all polynomials are define over field $\mathbb{F}_p$, where $p$ is a large prime number. Consider we have two polynomials, $p_1(x)$ and $p_2(x)$ (as defined above). For simplicity ...
0
votes
1answer
40 views

Find vectors x and y with given norms.

I've spent many hours on this and I just can't understand how to do this. Could you please go through this with me? I have a test, and I really need to understand how to do these types of problems. ...
0
votes
1answer
58 views

Solving linear equations over rings

Assume $$AX=0$$ is a linear system where $A$ is $n\times n$ matrix over ring $\Bbb r$ and $X$ is a length $n$ vector of variables if $rank_{\Bbb r}(A)=n-1$ how do we solve for the unique solution up ...
1
vote
1answer
29 views

On degenerate linear systems

Is it true that if in $$AX=0$$ where $A$ is $n\times n$ matrix over field $\Bbb K$ and $X$ is a length $n$ vector of variables if $rank(A)=n-1$ we will have an unique solution up to constant factors? ...
0
votes
1answer
60 views

Computing the Jordan Form of a Matrix

I apologize if this has already been answered, but I've seen multiple examples of how to compute Jordan Canonical Forms of a matrix, and I still don't really get it. Could someone help me out with ...
0
votes
1answer
41 views

how to find if the same matrix has one solution, infinitely many solutions and no solution

The matrix is as follows: $$ \left[ \begin{array}{ccc|cc} a&0&b&2\\ a&a&4&4\\ 0&a&2&b \end{array} \right] $$ I need to find when this matrix ...
3
votes
1answer
78 views

Why is this matrix skew-symmetric?

Consider Bilinear Forms, and two real $n \times n$ matrices $B$ and $\tilde B$. Suppose we have that $$(Bx,x) = (\tilde B x, x)$$ for all $x\in \mathbb R^n$. What can we can say about $B-\tilde ...
0
votes
1answer
66 views

Find the set of real values of $p$ for which the equation $ | 2x + 3 | + | 2x - 3 | = px + 6 $ has more than two solutions .

Options : A ) ( 4 , 0 ) B) R-{ 4 , 0 , -4 } C) {0} D) None How to find p = 0 , without having to analyse the graph ?
2
votes
1answer
47 views

Generalizing the norm and trace of finite extensions over finite fields.

I'm currently reading through Ireland and Rosen's A Classical Introduction to Modern Number Theory, and I'm working on proving that a later definition of trace and norm of arbitrary finite algebraic ...
1
vote
3answers
70 views

Prove that if rank $(\begin{smallmatrix} A &B\\ C&D\end{smallmatrix})=\operatorname{rank}(A)$, then $D=CA^{-1}B$ . [closed]

Let $A$ be an invertible $n \times n$ matrix with entries from a field $F$. Prove that if rank $\begin{pmatrix} A &B\\ C&D\end{pmatrix}=\operatorname{rank}(A)$, then $D=CA^{-1}B$ .
1
vote
1answer
25 views

Prove that there exists an orthogonal linear operator T on V such that $T\alpha_i=\beta_i$.

In an n-dimensional Euclidean space V two bases $\{\alpha_1,\alpha_2,…,\alpha_n\}$ and $\{\beta_1,\beta_2,…,\beta_n\}$ are given so that $(\alpha_i │\alpha_j)=(\beta_i│\beta_j)$ for all i and j. Prove ...
0
votes
0answers
23 views

Prove that the following conditions are equivalent: rank(A+B)=rank A + rank B; $R_A ∩ R_B = \{0\}$; $C_A ∩ C_B = \{0\}$.

Let the sizes of matrices A and B be equal, and let $R_A$ and $R_B$ be the spaces spanned by the rows of A and B respectively; let $C_A$ and $C_B$ be the spaces spanned by the columns of A and B ...
0
votes
1answer
19 views

Solving symbolic algebra matrix equations

Explicit X from the following equation: $[adj(A)\cdot A - I](A - X)^{T}=I|A|$ Considering that $|A|\neq 1$. A is an invertible matrix and $|A|\neq 1$ , show that: $[adj(A) - A^{-1}]^{-1} = \...
0
votes
1answer
33 views

finding eigenvalues and vectors for a linear transformation over a infinite dimensional Vector spaces.

let $V$={$(x_1, x_2,...,x_n...)$}|$x_i$ are real numbers } under normal operations and $T((x_1, x_2,...,x_n...))=(x_1+x_2,x_2+x_3,...,(x_n)+(x_{n+1})...)$ Find $$T((x_1, x_2...,x_n,...))=\lambda((...
1
vote
1answer
49 views

Minimal polynomial of a matrix of matrices.

Let the minimal polynomial of a matrix A be equal to $f(x)=(x-c_1)^{n_1}…(x-c_k)^{n_k}$. Prove that the minimal polynomial of the matrix $\begin{pmatrix}A & I \\0 & A \\\end{pmatrix}$ is equal ...
0
votes
1answer
21 views

How to show that the functionals given form a basis for the dual space.

I've done part $(a)$, but not sure about part $(b)$. If I understand the quetion correctly, then if $f$ is just a general polynomial of degree less than or equal to $n$, then $E_{x_0},\ldots,E_{x_n}$ ...
1
vote
1answer
17 views

Diagonalizability of a Linear Map

Please could somebody verify this: $T$ is diagonalizable $\iff$ there exists a basis of $V$ (the vector space ) consisting of eigenvectors of $T$ $\iff$ the algebraic multiplicity of each eigenvalue ...
-1
votes
2answers
51 views

Prove if $T : V\to U$ is linear, then the restriction$ T : \ker$⊥ $T \to$ range($T$) is one-to-one.

Can anyone prove this or point me in the right direction? If T : V → U is linear, then the restriction T : ker⊥ T → range(T) is one-to-one.
0
votes
0answers
18 views

Let $M\in M_n$ ,what is $W(M)$? [duplicate]

Define: Let $M\in M_n$ the numerical range of $M$ is $$W(M) = \left\{ {\frac{{{x^*}Mx}}{{{x^*}x}}:x \in {C^n},x \ne 0} \right\}$$ Now let $M = \left[ {\begin{array}{*{20}{c}} 0 & 0 & ...
4
votes
2answers
86 views

How can one show the inequality $\frac{a^2}{\sqrt{b^2-bc+c^2}}+\frac{b^2}{\sqrt{a^2-ac+c^2}}+\frac{c^2}{\sqrt{a^2-ab+b^2}}\ge a+b+c$

How can one show the inequality $$\frac{a^2}{\sqrt{b^2-bc+c^2}}+\frac{b^2}{\sqrt{a^2-ac+c^2}}+\frac{c^2}{\sqrt{a^2-ab+b^2}}\ge a+b+c$$ Where $a,b,c$ are real and $ab+bc+ac$ is no equal to zero This ...
3
votes
1answer
70 views

Number of variables equal to 0 in a homogeneous system of equations

For an $m\times n$ matrix $A$, let $k$ be the number of variables $x_i$ in $\vec{x}=(x_1,x_2,\dots,x_n)$ for which $x_i$ must equal $0$ in the solution to $A\vec{x}=\vec{0}$. For instance the ...