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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1answer
30 views

Solving according to complex numbers [closed]

If $n$ is an even natural number show that $(-1)^{n/2} =i^n$.
4
votes
1answer
32 views

The information the kernel provides about a linear map

In a linear map, if we know the mapping of the basis, we know all the information about the map. On the other hand, if we know the $\ker(f)$, $X/\ker(f)$ is isomorphic to $\mathrm{img}(f)$. I am ...
3
votes
1answer
49 views

Dimension of the image of a linear transformation

$$A=\begin{bmatrix} 0 &0& *& *\\ 0 &0& *& *\\ 0 &0& 0& *\\ 0 &0& 0& 0 \end{bmatrix}$$ (* denotes a non-zero element) If $A$ is considered as a linear ...
0
votes
2answers
31 views

Components of a vector given three points?

I'm trying to begin these two questions (25a and 26a specifically) but am at a loss on where to begin: No information on points on the graphs is given. I know the components of a vector with start- ...
-1
votes
1answer
26 views

Are trigonometric functions in the span of the collection $\{x^n\}$ of monomials? [closed]

Do trigonometric functions belong to span of $\{x^n\}$? I consider the answer to be yes, since all trigonometric function can be expressed in terms of powers of $x$.
2
votes
0answers
44 views

If one expresses a function as a linear combination of other functions, can the linear combination relationship be inverted?

If one has a function say f, that can be expressed as a linear combination of another type of function say g, can one invert the relationship as a linear combination of the other function? i.e. if one ...
3
votes
2answers
41 views

Spanning set is closed.

Suppose $\{e_1,e_2,\ldots,e_n\}$ is an orthonormal set in $\mathscr{H}$ (Hilbert space) and define $$M \equiv \operatorname{span}\{e_1,e_2,\ldots,e_n\}.$$ Show that $M$ is closed. Can I show that ...
3
votes
2answers
34 views

Finding the basis of a subset of polynomials

the question is as follows: Let $W$ be a subspace of the polynomials with maximum degree of $3$ and $p(1) = p(2) = 0$. Find the basis and the dimension of the subspace. The field is the real ...
2
votes
3answers
17 views

Direct Proof for Statement on Linear Independence and Unique Representations

The Statement Show that if a set of vectors is linearly independent, then any vector in the span of that set has a unique representation as a linear combination of these vectors. My Proof I'm going ...
1
vote
1answer
40 views

Determinant of a matrix with binomial coefficients.

Let $n \in\mathbb{N}$ and $A=(a_{ij})$ where \begin{equation}a_{ij}=\binom{i+j}{i}\end{equation} for $0\leq i,j \leq n$. Show that $A$ has an inverse and that every element of $A^{-1}$ is an integer. ...
1
vote
2answers
16 views

rotating linear dependent vectors in space

I'm not quite sure how to write this succinctly with mathematical symbols, so I just had to write it out in english. Any edit to suggest how to write it in mathematical form would be appreciated even ...
2
votes
1answer
56 views

What does $(G:G')$ mean?

I'm trying to teach myself some ring theory from a book, and have come across this sentence: "There are $(G:G') = 4$ linear characters" where $G$ is a group, and $G'$ is the derived group. I ...
3
votes
0answers
52 views

Resultant of two polynomials in two variables

I have two polynomials in two variables. $$f= nx^n+(n-1)x^{n-1}y+(n-2)x^{n-2}y^2+...+xy^{n-1}-c$$ $$g= x^{n-1}y+2x^{n-2}y^2+3x^{n-3}y^3+..+(n-1)xy^{n-1}+ny^n-d$$ Where $c$ and $d$ are some ...
1
vote
0answers
22 views

Any relation between Kernel methods and Variational methods?

I am familiar with kernel method, which is defined in the link: https://en.wikipedia.org/wiki/Kernel_method On the other hand I am familiar with variational methods which is defined in the link: ...
1
vote
1answer
27 views

A variant of the Vandermonde determinant

A very hard proof that $\sum_{i=0}^n i = \frac{n(n+1)}2$ (in comparison with the elementary level of the identity) is to compare degrees in the Vandermonde identity, which you prove playing around ...
-1
votes
2answers
53 views

Every combination of $v = (1, -2, 1)$ and $w = (0, 1, -1)$ has components that add to ________.

I am solving some problems in Linear Algebra text book. The question is filling in the blank of the following sentence. Every combination of $v = (1, -2, 1)$ and $w = (0, 1, -1)$ has components ...
1
vote
1answer
54 views

Positive semi definiteness of complex matrices [CSIR 2015]

Consider the following subsets of the complex plane: $$ \Omega_1=\left\{ C\in \mathbb{C} : \begin{bmatrix} 1 & C\\ \overline C &1 \end{bmatrix}\text{ is positive ...
0
votes
2answers
29 views

Is there always an injective linear functional? ($L^*$ surjective $\Rightarrow L$ injective)

I'm struggling with the (seemingly straightforward) linear algebra problem that follows: Let $V$ and $W$ be finite dimensional $\mathbb{ F } $-vector spaces and $L : V \to W$ a linear map. Define the ...
4
votes
1answer
36 views

question on self adjoint operator [duplicate]

Suppose $A$ is a $n\times n$ matrix with complex entries and $A^*A=A^2$. Does it imply $A=A^*.$
0
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0answers
20 views

Proving determinant of Vandermonde matrix

Problem: A matrix of the form \begin{align*} A= \begin{pmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ 1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\ \vdots \\ 1 & x_n ...
0
votes
1answer
16 views

Finding point on a line a given distance from another point on that line, 3D

I have two coordinates: $A(0.25, 0.0337, 0.5)$ and $B(0.3912, 0.1558, 0.3796)$ I would like to find the coordinates of a point $C$ along that line that is 0.1014 away from point $B$, but not between ...
-1
votes
1answer
28 views

how to prove the Wronskian?

I couldn't understand why it says the linear system has a nontrivial solution since the number of rows is :1(the original function)+(n-1)(take the derivative n-1 times)=n. And there are n columns or ...
0
votes
0answers
31 views

GRE Math Subject Test ,Please Check my Plan

I am to give test in october. I have planned to use Schaum series 3000 solved problems in calculus and 3000 solved problems in Linear Algebra for that .Also i will use Herstein for group theory .Since ...
4
votes
1answer
27 views

Differential Equation with Cross Products [without separating into system of equations]

I need to solve the following equation: $$ \frac{d m}{d t}=-m\wedge b-\alpha m\wedge (m\wedge b), $$ where $b$ is constant However, I was instructed specifically not to separate the calculation into ...
1
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1answer
27 views

Vector notation and tangent planes!

In my calculus textbook there is a part explaining tangent planes to surfaces, and how to form the normal vector for such a plane. I get the idea (taking the cross product of the tangential vectors) ...
0
votes
1answer
12 views

Transforming a tuple basis into a system of equations

Let $U=sp\{(1,4,1),(1,16,13) \}=sp\{u_1,u_2\}$. Find a condition (equation or a system of equations) on $a,b,c\in \mathbb R$ such that $(a,b,c)\in U$. In other words, show $U$ as a solution ...
0
votes
0answers
20 views

Invex function? How can I show?

Let $\mathbb{S}^m_+$ and $\mathbb{M}^{(m,n)}$, respectively, be the closed cone of positive semidefinite matrices and space of $m\times n$-dimensional matrices. Define function $F$ as ...
2
votes
1answer
43 views

The matrix $A-I$ is invertible, suppose $A^2=A$ and show that $A=0$

Let $A_{n\times n}$ over $F$ such that the matrix $A-I$ is invertible. Suppose $A^2=A$, prove that $A=0$. It's easy to see that $A(A-I)=0=(A-I)A$ I also know that there exist a matrix $B$ such ...
4
votes
2answers
38 views

How do I find $\|T\|$ when given a matrix $T$?

How do I find the norm $\|T\|$ of T: $\mathbb{R}^3 \rightarrow \mathbb{R}^3$ is defined by $T(x) := Ax$, where $A:= \begin{pmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 3 ...
0
votes
1answer
18 views

About finding intersection between two vector spaces

Let $W=sp \{e_1,e_2,e_3,e_4\}, U= sp\{(1,-2,1,0),(0,3,-1,1)\}$ be vector spaces both are linearly independent. Show that $U\cap W = sp\{(3,0,1,2)\}$. I know that $\dim U\cap W =1$. Now ...
0
votes
1answer
40 views

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$ I was trying to look at Hessian matrix and use Sylwester theorem, but I see that ...
0
votes
1answer
19 views

Geometric representation of principal axis

I would like to sketch $Q(X)=5x_1^2-4x_1x_2+5x_2^2=48$ So matrix $A=\begin{bmatrix} 5 & -2 \\ -2 & 5 \end{bmatrix}$ The eigenvectors are $\lambda=3, \lambda=7$. So $Q(X)=3y_1^2+7y_1^2=48$, ...
1
vote
1answer
27 views

proof that a cycle space is a subspace

I'm looking at the following proof that the cycle space of a graph is indeed a subspace, which I don't believe to be correct. proof: It suffices to prove that $\mathcal{C}$ is closed under $+$ ...
0
votes
1answer
39 views

Disjoint matrix multiplication

I'm studying matrix product algorithms. I've seen that there is a concept of disjoint matrix multiplication. What does it consist in? Thank you.
1
vote
1answer
24 views

Infinite matrix is injective if all its upper left minors are invertible?

This is a natural generalization of a recent MSE question. Let $X=(x_k)_{k\geq 1}$ be a sequence of real numbers, and $A=(a_{ij})_{i\geq 1,j\geq 1}$ be a real infinite matrix indexed by ${\mathbb ...
0
votes
1answer
13 views

Subvector and related subspace

This might be easier than I think, but I got stuck. Assume a vector $y=[y_1,\ldots,y_n]\in Y$, where $Y$ is a convex polyhedron. Assume a $k$-dimensional subvector of $y$, namely ...
1
vote
1answer
35 views

problem on Matrices and harmonic function [closed]

My Problem is: Given a matrix A of order 3 with real entries such that whenver $u(x)$ is a harmonic function of $(x_1,x_2,x_3)$ the function $v(x)=u(Ax)$ is also harmonic. then A can be: (a) Any 3 ...
-2
votes
1answer
46 views

If a $n \times n$ complex matrix $A$ satisfies $A^k=I_n$ and does not have eigenvalue $1$, then which of the following are necessarily true… [closed]

An $n\times n$ complex matrix $A$ satisfies $A^k=I_n$ where $k> 1$. Suppose $1$ is not an eigenvalue of $A$. Then which of the following are necessarily true- $A$ is diagonalizable. $A+A^2 \dots ...
0
votes
1answer
99 views

updating of the cholesky decomposition

I try cholrank1 update (wikipedia) of the symmetric positive definite (SPD) matrix . ...
0
votes
2answers
114 views

Prove a closed ball in $\mathbb R^3$ has an infinite number of extreme points.

How do I show that a closed ball in $\mathbb R^3$ has an infinite number of extreme points ? (Closed ball is written as $S = \{(x,y,z) \in \mathbb R^3 | \sqrt {x^2 + y^2 + z^2} \le R \}$) I know ...
0
votes
1answer
38 views

Choose the correct statement

Let $A$ be any $m \times n$ matrix of rank $n$ with real entries. Then choose the correct statement- $Ax=b$ has a solution for any $b$. $Ax=0$ does not have a solution. If $Ax=b$ has a solution, ...
0
votes
0answers
16 views

Does unitary transformation preserves the max of norm-2 of lower dimensional vectors?

Let $\{\mathbf{e}_i\in\mathbb{R}^n, i=1,...,N\}$. Apply a unitary transformation of the form $\mathbf{U}_N\otimes\mathbf{I}_n$ to this vector set and reach to vector set ...
2
votes
0answers
20 views

Vectorization of matrix (non-zero) entries

I am familiar with the $\textrm{vec}(A)$ operator, where the columns of a matrix $A$ are stacked into a vector. If my matrix $A$ has zero-entries, is there a standard notation/operator like ...
1
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0answers
18 views

Why $\|X-F\|=e|(X-F)\cdot N -d|$ should be written as $\|X-F\|=e|(X-F)\cdot N +d|$?

I'm reading Apostol's Calculus. $\quad $ And I've tried to do the following exercise: $\quad \quad \quad \quad $ I am a little confused: I have the portuguese version of the book, and it ...
4
votes
1answer
46 views

Invertible matrix of non-square matrix?

Is a matrix invertible only when it is a square matrix? What about a matrix of the order $m \cdot n$ with $m \gt n$ and such that it is row-equivalent to a row-reduced echelon matrix with more ...
1
vote
1answer
29 views

Let $u = (2, 3, 1)$, $v = (1, 3, 0)$, and $w = (2, -3, 3)$

I am struggling with this. Any help is appreciated. Let $u = (2, 3, 1)$, $v = (1, 3, 0)$, and $w = (2, -3, 3)$. Since $(1/2)u - (2/3)v - (1/6)w = (0, 0, 0)$ can we conclude that the set $\{u, v, ...
3
votes
1answer
51 views

Positive-definite function and Positive-definite matrix

I am trying to understand Positive-definite function and read the wikipedia link: https://en.wikipedia.org/wiki/Positive-definite_function It has a relation to Positive-definite matrix and I did not ...
1
vote
1answer
29 views

What happens if the power method is applied with a starting vector $q=c_2 v_2+…+c_n v_n$ in the presence of roundoff errors?

Supose $\{v_1,...,v_n\}$ is an eigenvector basis and $|\lambda_1|>|\lambda_2|>\ldots >|\lambda_n|>0$, so, my question is, if our starting vector $q \in span\{v_2,\ldots,v_n\}$ and in the ...
1
vote
1answer
45 views

Proof about dimension of subspaces

I'm working my way through Strang's Linear Algebra text, and I'm starting to get tripped up right at the end of chapter 3.5. I think I got 43, but I don't really understand 44 or 46. 43 ...
0
votes
0answers
16 views

Entry-wise norms of matrices

What is the use of entry-wise p-norm for $p>2$. I understand that $p=1$ and $p=2$ could be used as upper-bounds for maximum eigenvalue or spectral norm of positive matrices at least. But what ...