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

0
votes
0answers
5 views

About polynomials. If there is not two plynomials with the same grad in S, then S is linearly independent.

The problem states as follows. Let S be a set of polynomials non zero over a field F. If there is not two plynomials with the same grad in S, then S is linearly independent. I tried the following. ...
2
votes
0answers
8 views

using the finite field method to compute the characteristic polynomial of a hyperplane arangement

The following problem comes from Richard Stanley's $\textit{Enumerative Combinatorics}$ vol. 1, 2nd ed. It is problem 114 (c) in Chapter 3. Let $\mathcal{A}$ be a hyperplane arrangement in ...
1
vote
2answers
10 views

Eigenvalues and eigenvectors for orthogonal projection

I've been self-teaching myself linear algebra using Treil's Linear Algebra Done Wrong and I'm currently stumped on a problem and not sure how to start it. Here is the problem: If someone could give ...
0
votes
0answers
7 views

Area of a parallelogram with three dimensional vectors

There is a parallelogram that has the vertices 0, a, b, and a+b, all of which are three dimensional vectors. a = \begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}b = \begin{pmatrix} -1 \\ -2 \\ 0 ...
2
votes
2answers
341 views

Distance between two lines by orthogonal projection

I've got the lines' points and vectors $p,q$. My idea was to find a subspace (plane) with the basis of $p,q$ - perpendicular to the lines' axis. Then find the intersecting point $P$ of the lines' ...
0
votes
0answers
4 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
1
vote
1answer
23 views

How can I translate this problem into Matrix/Linear Algebra notation?

I have a matrix H of size s×d with holdings of s stocks across d days. H shows how many shares of each stock is in my portfolio on each day. The number of shares can change from day to day due to ...
1
vote
3answers
31 views

Square Root Confusion

well we know that $$\sqrt{x^2} = \pm x$$ Then if $$x^2=y^2$$ then $$\pm x= \pm y$$ Does this mean $x = y$ or $-x = -y$ or $x = -y$ or $-x = y$ or all is true? Which is true among these?
0
votes
2answers
17 views

subspace and subspace perp dimensions equal to V

I'm trying to prove the following statement: if E is a subspace of V, then dim E + dim $E^{\perp}$ = dim V. I know this is true because when these two subspaces are added, they are equal to V, but I'm ...
4
votes
0answers
119 views
+50

Limit of the projection of a matrix when the projection is not continuous

Consider two real matrices: the $n\times n$ matrix $A$ the $n\times m$ matrix $B$ of rank $m$, with $m<n$. Let, for $a\in\mathbb{R}$, $$S_a=A-aI_n,$$ and denote by $P_a$ the orthogonal ...
-1
votes
0answers
11 views

Linear Programming problem

Find values of the variables x1, x2,, and x3 which satisfy x1 + 2x2 + x3 ≤ 16 4x1 + x2 + 3x3 ≤ 30 x1 + 4x2 + 5x3 ≤ 40 so that the minimum value of x1, x2, and x3 is as large as possible. Write this ...
2
votes
1answer
45 views

If $A$ is positive definite then so is $A^k$

Prove that if $A$ is positive definite, then so are $A^2,A^3,\ldots$ and $A^{-1},A^{-2},\ldots$ I know how to show the inverse of positive definite is positive definite but I don't know how to expand ...
0
votes
0answers
21 views

If zero is an eigenvalue of the matrix A, What does it say about A? [on hold]

An eigenvector cannot be zero, but an eigenvalue can. Suppose that zero is an eigenvalue of A. What does it say about A? Hint: One of the most important properties of a matrix is whether or not it is ...
0
votes
1answer
12 views

Given a survival rate matrix, describe what can be said about it

Given this matrix equation: $$\begin{bmatrix} c_{k+1} \\ t_{k+1} \\ a_{k+1} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0.33 \\ 0.18 ...
1
vote
0answers
15 views

Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
1
vote
0answers
24 views

finding column vectors - linear transformations

$L:\mathbb{R}^3\rightarrow \mathbb{R}^2$ with bases $\mathcal{S}=\left\{\left(-1,1,0\right),\left(0,1,1\right),\left(1,0,0\right)\right\} \: \text{for} \:\mathbb{R}^3 \:\text{and} \\ ...
0
votes
0answers
8 views

Diagonal of multidimensional DFT

If $X$ is a $n\times n$ square matrix and $F$ its Discrete Fourier Transform, is there a way to compute the diagonal $(F_{1,1},\ldots,F_{n,n})$ without explicitly computing the full DFT? How about ...
1
vote
1answer
25 views

Gauss-Newton Non-Linear Squares Optimisation

I doubt this is solvable at all, but I thought I will give a try. Essentially I am trying to extend Gauss-Newton algorithm to 2nd Taylor term. ...
1
vote
0answers
8 views

Inverse properties of $L_1$ normed matrices

Let's take the adjacence matrix $A$ of a graph $G$. We call $\bar{A}$ the row $L_1$ normalized matrix obtained from $A$. Let's take some $\alpha \epsilon [0,1)$. $(I-\alpha\bar{A})$ is strongly ...
1
vote
0answers
18 views

linear transformation of orthogonal vector space on subspace

Let $V$ be a finite dimensional inner product space over $F$. If $W$ is a subspace of $V$, prove that the orthogonal projection of $V$ on $W$ is an idempotent linear transformation of $V$ into $W$. I ...
0
votes
1answer
22 views

How to count algebraic multiplicities to show $\nexists$ an eigenbasis for $A$?

If $A=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix},f_A(\lambda)=(1-\lambda)^3 \,\text{and } E_1=\text{ker ...
2
votes
1answer
20 views

Solution for system of quadratic equations

Can anyone provide a straightforward solution to the following equation: $\vec{y}=M\vec{x}+N\vec{x^2}$ where $\vec{x^2}$ is a column vector with each component being the squared value of the ...
1
vote
1answer
21 views

Calculating determinant matrix with size of n

we got the following matrix in order of $n$x$n$: $$\begin{pmatrix} 1 & 0 & . & . & . & 0 & 1\\ 1 & 1 & 0 & . & . & . & 0\\ 0 & 1 & 1 & 0 ...
-6
votes
1answer
41 views

Dont ask - What is the relation between $f$ and $f(x)$? [on hold]

$f = a.g + b.h$, space $V$ $f(x) = a.g(x) + b.h(x)$ $f$, $g$ and $h$ are scalar valued functions. $x$ is a vector in $\Bbb R^1$ and $f$ is a vector in $V$. So $[f(x)]$ is a vector. is $[f(x)] (f) ...
1
vote
0answers
18 views

Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
-1
votes
1answer
14 views

Domain of compostions of linear mappings

Let $T$ be a linear transformation from $\Bbb R^3$ into $\Bbb R^2$ and $S$ be a linear transformation from $\Bbb R^2$ into $\Bbb R^3$. Is the mapping $ST$ a linear transformation from $\Bbb R^3$ into ...
1
vote
0answers
34 views

Does this linear system of 5 unknowns and 2 equations have multiple solutions? [on hold]

\begin{cases} x+ 2y - z + w - t = 0 \\ x - y + z + 3w - 2t = 0 \end{cases} Add 1st to the 2nd: $$2x + y + 4w - 3t = 0 \\ y = -2x - 4w + 3t = 0$$ Substitute y in the 1st: $$x - 4x - 8w + 6t - z ...
0
votes
2answers
38 views

How do you solve this circular system of equations in $\mathbb{Z}_2$?

I'm trying to solve a system of equations in $\mathbb{Z}_2$ that look like this: \begin{align} x_1 + x_2 = p_1 \\ x_2 + x_3 = p_2 \\ x_3 + x_4 = p_3 \\ ... \\ x_n + x_1 = p_n \\ \end{align} I know ...
1
vote
2answers
17 views

How to find matrix of orthogonal projection from gram-schmidt orthogonalization

I'm having a little difficulty understanding Gram-Schmidt orthogonalization. I have a problem to apply Gram-Schmidt orthogonalization to the system of vectors $(1,1,1)^T, (1,2,1)^T$ then write the ...
1
vote
1answer
13 views

Showing something involving integrals is an inner product

I have this problem: Let $C([0,1])$ be the real vector space of continuous functions on the interval [0,1]. Show that $<. , .>: C([0,1]) \times C([0,1]) \rightarrow \mathbb{R}$ ...
1
vote
1answer
15 views

What the limit of a matrix over time shows about the future

$x_k$ is the fraction of people who prefer cake to pie at year $k$. The remaining fraction $y_k=1-x_k$ prefer pie. At year $k+1$, $\frac{1}{5}$ of those who prefer cake change their mind. Also at year ...
0
votes
2answers
326 views

proof of the full exchange lemma

Let V be spanned by $\{v_1,...,v_k\}$ and let $\{u_1,...,u_k\}$ be a linearly independent subset of V, then: 1) $k\leq n$ 2) $\exists$ a spanning set $\{w_1,...,w_n\}$ for V where $w_i = u_i$ for $1 ...
1
vote
2answers
34 views

Matrix with eigenvalue that should equal 1.

I have the matrix: $$A = \begin{bmatrix}4 & -2 & 3\\0 & -1 & 3\\-1 & 2 & -2 \end{bmatrix}$$ and I need to find out if $\lambda = 1$ is an eigenvalue. So I solved the equation ...
0
votes
0answers
18 views

A thought about transition matricies in vector spaces

I am trying to work out the relationship between transformation matricies of a vector space with different bases. I came up with an equation which does not look right, but I would like your opinion. ...
0
votes
1answer
46 views

about derivative of a matrix and trace

I have checked it up the following derivation of a formula:" The question that I have is why the author uses the trace in the third part; supposedly it uses a formula derived from the properties of ...
2
votes
1answer
37 views

Characteristic polynomial of a matrix polynomial

Thanks for any help or comments. Suppose $A\in M_n(F)$ is an $n\times n$ matrix such that $F$ is a finite field. Also suppose that characteristic polyomial of $A$ is irreducible and is equal to its ...
0
votes
0answers
9 views

Congruence Property of Monotone Operators

Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if $T:\mathbb R^n\rightrightarrows\mathbb R^m$ is strictly monotone and $\text{rank}\;A=n$, then $S:=A^TT(Ax+b)$ is also ...
1
vote
4answers
41 views

A real $2 \times 2 $ matrix $M$ such that $M^2 = \tiny \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$ , then :

A real $2 \times 2 $ matrix $M$ such that $$M^2 = \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$$ (a) exists for all $\epsilon > 0$. (b) does not exist for any ...
1
vote
1answer
22 views

Different Inverse Approach

As it is known, we use inverse (Gauss Elm, Jordan...) or pseudo-inverse methods (LU, SVD, Chol, QR...) to solve linear equation namely $ A*x=b$ when $A$ is $[m,n]$ and $b$ is $[1,n]$ matrix. These all ...
2
votes
1answer
23 views

How can I prove that any matrix A can be expressed as the sum of two Hermitian matrices , B and C, in the form A = B + iC?

The question is in the title really. Whether or not A must also be Hermitian is not clear to me. Sorry, I am not very good with proofs of this nature.
4
votes
1answer
494 views

Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known

Let $A$ be an $n \times n$ symmetric matrix of rank $n$ with known inverse $A^{-1}$. Let $D$ be a diagonal matrix with the same dimensions and rank. What is the fastest way to compute $(A+D)^{-1}$? ...
-1
votes
1answer
20 views

Proving multilinearity of determinant [on hold]

As the title says, how we can prove multilinearity property of determinants: $$ \begin{vmatrix} p+q+r & x+y+z & u+v+w \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\\ ...
1
vote
1answer
16 views

Finding orthonormal basis using orthogonal basis

I am very confused how to go about finding an orthonormal basis using a orthogonal basis. My book says to just normalize the vectors but it doesnt further explain. When i look at answers for ...
0
votes
3answers
48 views

For which values of a do the following vectors form a linearly independent set in R^3

I've seen this same question, but asking for linearly dependent, not linearly independent. $$ V_1= \left(a,\, \frac{-1}{2}, \,\frac{-1}{2}\right),\;\; V_2= \left(\frac{-1}{2},\, a, ...
0
votes
1answer
76 views

Linear Algebra, multiplication of Tensor by vector by vector.

I am deriving some equations and need to know the correct mathematical notation for opening up the brackets of an equation with the following variables: tensor $A \in$ ${\mathbb R}^{l \times l \times ...
0
votes
1answer
25 views

Condition of distinct eigenvectors?

I am looking at this wikipedia page http://en.wikipedia.org/wiki/Matrix_decomposition#Eigendecomposition ...
1
vote
5answers
88 views
+50

Proving rank of $AB$ is at most equal to rank of $B$

$A=m\times n$ matrix. $B = n\times p$ matrix. Prove that the rank of of the product $AB$ is at most equal to the rank of $B$. Current state of my work: (1) First idea: show that the kernel of $B$, ...
0
votes
3answers
34 views

Proof of linear independence of non-empty subsets

The question states: Show that if $S = \{v_1, v_2, \ldots , v_r\}$ is a linearly independent set of vectors, then so is every non-empty subset of $S$. I understand that if $r>n$, $S$ is ...
0
votes
2answers
47 views

Vector spaces whose elements are functions

I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. It seems that functions can be manipulated as vectors as ...
2
votes
1answer
38 views

Use determinants to calculate the area bounded by 3 vectors

I have seen the proof of why the area of the parallelogram created by 2 vectors $u = \left(\begin{matrix} u_1\\ u_2 \end{matrix}\right)$ and $v = \left(\begin{matrix}v_1 \\ v_2 \end{matrix}\right)$ ...