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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Is there a decomposition $U U^T$?

We know that there exist the Choleski decomposition $ M = L L^T $ where $M$ is a positive definite matrix and $L$ a lower triangular one. Does it exist a similar decomposition in $ M = U U^T$ ...
3
votes
1answer
72 views

Finding possible Characteristic and Minimal polynomials of a matrix

$A$ is a 3x3 matrix over $\mathbb{C}$ that's not diagonalizable with trace 3 and determinant 1. Instructions: Find all possible characteristic and minimal polynomials. Characteristic polynomials: ...
2
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2answers
106 views

Is the 4-norm of the orthogonal projection of a vector into a subspace less than or equal to the 4-norm of the vector itself?

To simplify the question, suppose $V\subseteq \mathbb{R}^n$ is a subspace and $P$ is the projection matrix. Is it true that $\forall x \in \mathbb{R}^n, \|Px\|_4 \le \|x\|_4$. I guess it is true for ...
0
votes
1answer
50 views

Does an orthonormal matrix preserve the $p$-norm?

Let $\,A\,$ be a $\,n \times k\, $ matrix, and $\,B\,$ a $\,k \times n \,$ be an orthonormal matrix. Is it true that $\,\left\|AB\right\|_p = \left\|A\right\|_p\,$ for every $\,p\neq 2$?
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51 views

cholrank1 update with LDL decomposition

I have cholrank1 update procedure (wikipedia) for the symmetric positive definite (SPD) matrix. ...
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53 views

Forward and Backward Substitution

Let L be a lower triangular matrix and solve Lx=b by forward substitution. Show that barring overflow or underflow, the computed solution x̂ satisfies (L +δL)x̂=b, where |δlij| ≤ ηε|lij|, where ε is ...
-2
votes
2answers
69 views

Span of two vectors in $\mathbb{R}^2$ [closed]

The span of two vectors in $\mathbb{R}^2$ neither of which is zero vector, and which are not parallel, is- a point. line in $\mathbb{R}^2$ not running through origin. line in $\mathbb{R}^2$ running ...
3
votes
2answers
45 views

How to prove the space of all linear transformations from $V$ to itself isomorphic to $M(n \times n, F)$?

Dimension of $V$ is $n$ here. I know that the two spaces have the same dimension. Does this suffice to say they're isomorphic. How so?
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2answers
38 views

Augmented matrix gives no solutions when augmented with variables, but will give solutions when augmented with some numbers

I have a homework problem that boils down to this: x1 = 1a + 1b x2 = 1b + 1c x3 = 1c + 1d x4 = 1d + 1e x5 = 1e + 1f x6 = 1f + 1a I write the matrix to solve for x. I row-reduce the matrix. ...
0
votes
1answer
16 views

Projective coordinate basis generated by a triangle and a fourth non-collinear point

I have read in several places, that give four points in the projective plane, with no three of them collinear, that you it is possible to create a linear change of coordinates such that the points can ...
0
votes
1answer
15 views

If $A$ is a positive definite bilinear form defined on a real vector space, does it satisfy the following condition?

If $A$ is a positive definite bilinear form defined on a real vector space, does $A$ satisfy $2A(u,u)^\frac 12 A(v,v)^\frac 12 \ge A(u,v) + A(v,u)$?
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2answers
53 views

Find a marix of a linear transformation given its effect on vectors

Im trying to find the matrix corresponding to this linear combination. I cant seem to get the right answer. $T(1,0,0)^t=(1,0,0)^t$ $T(-1/2,\sqrt{3}/2,0)^t=(-1/2,-\sqrt{3}/2,0)^t$ $T(-1/2,-\sqrt{3}/...
2
votes
1answer
43 views

Number Theory and Linear Equations

If the equation $$9x + 13y = K$$, has exactly five solutions, where x and y are positive integers, what is the minimum possible value of K? My work till now has been really simple : $$\frac{(K-13y)}{...
0
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1answer
41 views

Diagonalizability of real Normal matrix.

I am reading Hoffman and Kunze linear algebra book. In chapter $8$ there is a theorem that states as " Let $V$ be a finite-dimensional complex inner product space and $T$ a normal operator on $V.$ ...
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3answers
52 views

Eigenvalues of operator given polynomial

I have a homework question that is really bothering me because I cannot manage to use any of the theorems we have studied so far, it seems. Could someone please give me a pointer please (no, Cayley-...
2
votes
1answer
26 views

Algebra with complex numbers (i)

My professor has assigned this problem: Let r be a real number. By using r = r + 0i, confirm that r(a + bi) = (ra) + (rb)i. My understanding is that this looks at distributive property, but I do ...
2
votes
0answers
23 views

Explicit construction of a basis in a finite dimensional vector space [duplicate]

Consider a vector space $V$ of dimension $d$, and suppose you are given $d-1$ linearly independent vectors $v_1, \dots, v_{d-1}$. Is there a simple explicit expression for a vector $v_d$ which ...
0
votes
1answer
35 views

Is it true that a positive definite bilinear form on a real vector space is an inner product?

I don't think the property $\langle u,v \rangle = \langle v,u \rangle$ is satisfied, if the bilinear form is not symmetric. Could anyone help me?
0
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0answers
19 views

Geometric interpretation of an n-dimensional outer product.

For an n-dimensional vector $u$, the matrix $uu^T$ represents a projection onto the space (?) spanned by $u$. What is the action of the matrix $I - uu^T$ where $I$ is the identity matrix? Can ...
0
votes
1answer
31 views

Linear dependence/independence and finding all solutions to the linear system

Consider the matrix, $M = \begin{bmatrix} -4 &1 &1 \\ 2& 0&1 \\ 0&1 &3 \end{bmatrix}$. Are its columns linearly dependent or linearly independent vectors? Justify your ...
1
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1answer
31 views

Show that $H+K$ is a subspace of $V$

I am trying to solve this, please give me hints. $H,K \in V$, $V$ is a vector space. $H+K=\{w:w=u+v:u \in H, v \in K \}$ Show that $H+K$ is a subspace of $V$.
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votes
1answer
29 views

Largest number of elements that a basis for V in $R^n$could have?

Let $V ⊆ R^n$ be a subspace of $R^n$ (a) What does it mean for a set of vectors $S$ = {$v1, . . . , vk$} in $R^n$ to be a basis for V ? (b) What is the largest number of elements that a basis for V ...
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2answers
53 views

Matrix rank linear algebra [closed]

We consider two matrices $A, B \in M_4(\mathbb C)$, $a \in \mathbb{C}^*$ and $$AB + aA + aB = 0 .$$ How can I prove that $AB = BA$ and $\operatorname{rank}(A) = \operatorname{rank}(B)$?
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1answer
180 views

How do I solve the $Ax=B$ equation where $A$ is not a square matrix?

Let $A=\begin{bmatrix}2&-1&2&0\\-3&0&2&1\\-4&-1&-2&-1\end{bmatrix}$. I need to solve find $X$ in $AX=B$, where $B=\begin{bmatrix}1\\-1\\0\end{bmatrix}$. How am I ...
0
votes
2answers
31 views

Give an example in $\Bbb R^2$ to show that the union of two subspaces is not, in general, a subspace.

I'm trying to solve this on my own. Give an example in $\Bbb R^2$ to show that the union of two subspaces is not, in general, a subspace. I feel the need to use functions to illustrate this, but it ...
0
votes
1answer
53 views

Solving linear systems over $\mathbb{Z}/n$

I am given the following system of linear equations: $2x + 5y \equiv 6 $ $3x + 6y \equiv 5 $ and I am asked to solve it over the set $ \{ 0,1,2,3,4,5,6 \} $, that is, over $\mathbb{Z} / 7 \mathbb{Z}...
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0answers
38 views

Simple question on the linearity of a dynamical system

Take a continuous-time dynamical system $\Sigma=(\mathbb{T},\mathbb{W},\mathfrak{B})$ with $\mathbb{T}=\mathbb{R}$, $\mathbb{W}=\mathbb{R}$ and all sinusoidal signals with period $2\pi$. i.e. $w:\...
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vote
1answer
337 views

Proving that a symmetric matrix is positive definite iff all eigenvalues are positive

This has essentially been asked before here but I guess I need 50 reputation to comment. Also, here I have some questions of my own. My Proof outline: (forward direction/Necessary direction): Call ...
2
votes
2answers
34 views

Linear application in canonical basis

I have the function $f:\mathbb C^3 \to \mathbb C^3$, $f(x_1,x_2,x_3) = (4x_1 + 2x_2, x_1+4x_2+x_3, x_1+x_2+4x_3)^t$. $t$ stands for transposed How can I determine if it is an linear application and ...
1
vote
2answers
67 views

Does the determinant of a matrix made up of column vectors being non-zero imply that the vectors are independent?

Let's say we have 3 vectors and we make up a matrix where we depict the vectors as the columns of the matrix. If we calculate the determinant of the matrix and we get a non-zero number, does that mean ...
3
votes
1answer
39 views

Diophantine-like equations

So I was solving a problem and encountered a specific system of equations that I don't know if a solutions exists for it or not. $$\begin{align} 4ny&=d^2-a^2\\ -4nx+4ny&=d^2-b^2\\ 4nx&=d^...
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votes
4answers
41 views

Linear map of derivatives

Let $V$ be the vector field of all polynomials in $\mathbb{R}$ of degree at most $3$. If $T:V\rightarrow V$ is the linear map that sends a polynomial $p(x)$ to $p(x)+p''(x)$, what is the matrix that ...
2
votes
0answers
37 views

On similar matrices.

I am searching two matrices such that both of the matrices have same characteristic polynomial, same minimal polynomial, same rank, same trace, same determinant, same algebraic and geometric ...
0
votes
1answer
54 views

How does the definition of compactness imply that all continuous operators are compact in finite dimensional spaces?

Let $S \subset X, Y$ be normed spaces over $K$. An operator $A:S \to Y$ is called compact if: $A$ is continuous $A$ transforms bounded set into relatively compact sets i.e. if $(c_n)$ is ...
0
votes
1answer
20 views

Ratio of two quadratic vector forms

This is probably a very basic question. Suppose we have two $n \times n$ positive definite real matrices A and B, and $x \in \mathbb{R}^n$. What is the value of following ratio as a function of $x$? $...
1
vote
1answer
48 views

Computing Eigenvector.

I have to find the eigenvalues and the eigenvectors of the following matrix : $\begin{equation*} \mathbf{S} = \left( \begin{array} {cc} .0144 & .0117\\ .0117 & .01466 \end{array} \right) \end{...
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vote
0answers
40 views

Showing $P^n $ is isomorphic to $R^{n+1} $

Show $P^n$, the vector space of all polynomials of degree $\le n$ with real coefficients, is isomorphic to $\mathbb R^{n+1}$. I always have trouble showing that a function is one to one and onto. So ...
1
vote
2answers
83 views

Why would the eigenvalues of this type of (stochastic) matrix all be close to 1?

I'm working with matrices defined by $$T_{ki} = \sum_{j} M_{kj}N_{ji} + \delta_{ki}\biggl(1 - \sum_{j}N_{ji}\biggr)$$ where $M$ is a stochastic (or probability) matrix, where $\sum_{k} M_{kj} = 1$, $N$...
0
votes
1answer
44 views

Finding coordinates relative to basis (Linear algebra)

The set $V$ = {$(a + 2b + 3c, 3a + b + 4c, 4a + 3b + 7c) : a, b, c ∈ R$} forms a subspace of $R^3$ (a) Show that $S$ = {$(1, 3, 4),(2, 1, 3),(3, 4, 7)$} is a spanning set for $V$ . (b) Find a ...
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0answers
97 views

Prove the general linear group is open in $\mathbb{R}^{n^2}$

I was wondering if somebody could help me with the following problem. Let $GL(n,\mathbb{R})$ be the space of $n \times n$ matrices over $\mathbb{R}$. Prove that $GL(n,\mathbb{R})$ identified in the ...
2
votes
1answer
39 views

Eigenvalues of an orthogonal matrix over an arbitrary field

Suppose we have an orthogonal matrix $A$ (i.e. $AA^T = E$) over an arbitrary field $\mathbb k$ ($char$ $\mathbb k \ne 2$, algebraically closed if it is needed) and $\det A = -1$. Is it possible that ...
0
votes
1answer
41 views

Is it always possible to find a vector perpendicular to two given vectors in a general inner product space?

In short, given an inner product space $X$ , for any $x,y \in X$ does there always exist a nontrivial $z \in X$ so that $<x,z> = 0$ and $<y,z> = 0$ ?
3
votes
2answers
95 views

Finding an orthogonal basis w.r.t an inner product

I am trying to figure out a question: Let $W$ be the subspace $\mathbb{R^4}$ spanned by vectors $$v_1=(1,1,1,1), v_2=(1,0,1,0) \text{ and }v_3=(1,1,0,0)$$ equipped with the standard Euclidean ...
0
votes
1answer
110 views

hamming code using linear algebra

You have received the message 1110111 coded by (7,4)-Hamming. What 4-bit word did the sender want to convey? I know hamming code but I need to solve it using linear algebra. I know its a linear code ...
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1answer
47 views

Prove $\|x\|_1\le \sqrt n \|x\|_2$ [closed]

Prove $$\|x\|_2 \le \|x\|_1 \le \sqrt n\|x\|_2.$$ I already proved the first inequality $\|x\|_2 \le \|x\|_1$. Please help me with the second part: $$\|x\|_1\le \sqrt n \|x\|_2.$$
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0answers
29 views

Finding spanning vector sets

Let $V$ be the set of all vectors over the non-negative integers. For any two subsets $S$ and $T$ of $V$, define $S + T$ to include: All vectors in $S$ All vectors in $T$ All vectors that can be ...
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0answers
13 views

linearly independent lists

Verify that the list $1,z,...,z^m$ is linearly independent in $\mathcal{P}(F)$ for each non-negative integer m. $\mathcal{P}(F)$ is the set of all polynomials with coefficients in F. A list is ...
1
vote
2answers
57 views

How would I prove this is a subspace?

"Determine if the set $H$ of all matrices in the form $ \left[ \begin{array}{cc} a & b \\ 0 & d \\ \end{array} \right] $ is a subspace of $M_{2\times2}$." And I'm given, A subspace of a ...
5
votes
1answer
43 views

$SL_2(\mathbb{Z})$ a subgroup of $SL_2(\mathbb{R})$? [closed]

As the title suggests, is $SL_2(\mathbb{Z})$ a subgroup of $SL_2(\mathbb{R})$? Thanks in advance.