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
0answers
13 views

Number of fixed points of an orthogonal transformation

Question is as follows : Let $T:\mathbb{R}^3\rightarrow \mathbb{R}^3$ be an orthogonal transformation such that determinant of $T$ is $1$ and $T$ is not the identity linear transformation. Let ...
0
votes
1answer
17 views

Elements fail to form a basis

Consider the vector space $P$2 and the set $$5−1t+4t^2,−4+3t+1t^2,8+5t+kt^2$$ For which $k \in \mathbb{R}$, do these three elements fail to be a basis of $P$2? I thought in order to make the three ...
1
vote
1answer
30 views

Why does the additive inverse not follow

I need to prove that the vector space of $\mathbb{R}^2$ with the following operations: $x + y = (x_1 + 2y_1, 3x_2 - y_2)$ The usual scalar multiplication of $cx = (cx_1, cx_2)$ The answers in my ...
1
vote
1answer
22 views

Expressing linear transforms using linear functionals: is this possible?

Work over a fixed but arbitrary field. Let $Y$ and $X$ denote finite-dimensional vectorspaces, and let $y \in Y^n$ denote a sequence of elements of $Y$, where $n$ is a natural number. It seems ...
3
votes
2answers
37 views

Understanding Eigenvalues, Eigenfunctions and Eigenstates

Please could somebody explain the meaning and uses of Eigenvalues, eigenfunctions and eigenstates for me. I have taken 3 years of physics and math classes at university and never fully grasped the ...
1
vote
0answers
12 views

Eigenvalues of the subtraction of a gram matrix and a psd rank $1$ matrix.

If $V$ is $p \times r$ matrix ($r<p$) with rank $r$, then $G=V^TV$ is a $r \times r$ gram matrix with rank $r$. Let $x$ be a $r \times 1$ vector, and $xx^T$ is a psd rank $1$ matrix. It's already ...
1
vote
0answers
25 views

Attempt to solve a matrix (counterbalancing) problem computationally gives “spooky” result: why?

This question is posted on the mathematics section of stackexchange because my uneducated guess is that the answer involves some basic mathematical principles, possibly in the domain of linear ...
0
votes
1answer
17 views

Compute the matrix with the standard basis.

Let $X$ be the left representation of $S_3$. Compute the matrix $X(132)$ in the standard basis $S = {\{123,132,213,231,312,321}\}$. attempt: In general if $G = S_3 = {\{\pi_1, \pi_2,..,\pi_6}\}$, ...
1
vote
0answers
28 views

Linear Alg. Short proof on determinant

Hi can I get a quick check on my proof to see if it is correct. proof
-1
votes
0answers
18 views

If two linear systems are equivalent, they have the same size augmented matrix. [on hold]

If two linear systems are equivalent, they have the same size augmented matrix? It is false but do any one know why for this?
0
votes
1answer
15 views

Proving left inverse for $A$

Taken from Artin's book, need to prove that an $m \times n$ matrix $A$ (where $m < n$) has no left inverse. A hint given is to compare matrix $A$ to an $n \times n$ matrix obtained by taking $A$ ...
0
votes
0answers
12 views

What star rating is representative of this distribution?

100 people vote. They can vote 1, 2, 3, or 4 stars. Distribution: 1 = 33, 2 = 26, 3 = 12, and 4 = 28. What star rating would you say is "representative" of these 100 people: 2.36 (2), the average, ...
0
votes
0answers
11 views

Picture of Orthogonal Projection and Parallel Projection which is not a orthogonal projection in $3$ dimensions.

I would like to see a picture of an orthogonal projection in $3$ dimensions. I would also like to see a picture of a parallel projection that is not an orthogonal projection in $3$-dimensions. Does ...
1
vote
0answers
7 views

Eigenvalue ratio evolution of Laplacian matrix when add edges

Consider an connected digraph, we use the classic definition of the Laplacian matrix $L$: $L=D-A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. There has been many researches on ...
0
votes
1answer
11 views

Show that for each $v \in V$ exist $w \in W$ and $c \in \Bbb R$ unique such that $v=cv_0+v$

Let be $V$ a vector space over the field of real numbers, $f \in V^*$ and $W=ker (f)$. If $v_0 \in V$ is a vector such that $f(v_0)\neq0$, show that for each $v \in V$ exist $w \in W$ and $c \in \Bbb ...
2
votes
1answer
26 views

Proof of Cauchy-Schwarz Inequality 1

In my lecture notes I've written the proof of Cauchy-Schwarz inequality as: Let t $\in$ R and $\langle x+ty, x+ty\rangle \geq 0$, then $\langle x+ty, x+ty\rangle $ = $\langle x, x+ty \rangle + ...
2
votes
0answers
32 views

Supremum equal Max

Let $p$ be a polynomial and $\|.\|_A$ is a norm defined by $$\|\mathbf{x}\|_A:=\sqrt{\mathbf{x}.A\mathbf{x}},$$ for $\mathbf{x}\in\mathbb{R}^n$ and $A\in\mathbb{R}^{n\times n}$. Let $A$ be a symmetric ...
-2
votes
3answers
39 views

How do I determine if A is diagonalizable [on hold]

$A= \begin{bmatrix} -5 & 1 & 5 \\ -7 & 4 & 4 \\ -1 & 1 & 1 \end{bmatrix}$ How do I show that A is diagonalizable?
0
votes
2answers
20 views

How to find the limit of a matrix $P^n = UD^nU^{-1}$ where $D$ is a diagonal matrix of eigenvalues and $U$ a matrix of eigenvectors?

If we have a matrix where $P = UDU^{-1}$, where $D$ is a diagonal matrix of real eigenvalues that are less than or equal to 1, and $U$ is the corresponding matrix of eigenvectors, how can we show that ...
1
vote
2answers
26 views

Find a basis for Each corresponding eigenspaces

$A$= $\begin{bmatrix} -5 & 1 & 5 \\ -7 & 4 & 4 \\ -1 & 1 & 1 \end{bmatrix}$ I now want to find the eigenctvectors of $A$ and the basis corresponding to each eigenspaces. ...
1
vote
1answer
12 views

Row Equivalent Matrices

If I have a matrix $A$, where there are zeros everywhere apart from the first row, what are the matrices that are not row equivalent to $A$. I know that if two matrices are row equivalent, we can ...
0
votes
2answers
22 views

Understanding basis algorithm result

I've a matrix ${\bf A}$ defined as A = \begin{pmatrix} 1 & -2 & 0 & 3 & 7\\ 2 & 1 & -3 & 1 & 1\\ \end{pmatrix} And ${W_1}$ is the solution ...
1
vote
3answers
28 views

Matrices and diagonalization.

I could verify that $P$ statement is false by just calculating the determinant but couldn't answer $Q$ statement. Any clue about $Q$??
0
votes
3answers
36 views

find when matrix is not diagonalizable

Let $A = \begin{pmatrix} 3&0&0\\ 0&a&a-2\\ 0&-2&0 \end{pmatrix}$ A is not diagonalizable find $a$. how can I tell when $a$ is diagonalizable by it's characteristic ...
0
votes
2answers
32 views

It is true that $rank(xy^T)=1$? [on hold]

Let $x,y\in \mathbb{C}^n$. It is true that $rank(xy^T)=1$?
0
votes
0answers
10 views

Piecewise linear function given three points and two crossover boundaries

Suppose you have three points; $(3500, 700)$, $(52500, 5075)$, and $(527500, 36800)$. As well as two $x$ boundaries $25000$ and $200000$. The question is then to construct three lines (each of which ...
2
votes
1answer
24 views

Finite abelian group as Z-module

If $M$ is a finite abelian group then $M$ is naturally a $Z$-module. Can this action be extended to make $M$ into a $Q$-module ?
0
votes
1answer
26 views

Linear trasnformation kernel and image [on hold]

$V$ is a vector space. Let $T: V \to V$ be a linear transformation. Prove that if $\text{Ker}\: T = \text{Ker}\: T^2$ then $\text{Im}\:T = \text{Im}\:T^2$. How do I prove it?
1
vote
0answers
10 views

Is it possible to write the Hadamard product of two matrices in tensor notation?

Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common ...
-1
votes
1answer
33 views

Image of a linear transformation

Let $T : V \to W$ be a linear transformation. If $A$ is a subspace of $V$, show that its image, $$ T(A) = \left\{ T(x) \in W \mid x \in A \right\}, $$ is a subspace of $W$. I have no idea how ...
-2
votes
1answer
13 views

To find dimension of $N(A) \cap R(B)$ over R [on hold]

To find dimension of $N(A) \cap R(B)$ over R A = $\begin{bmatrix} 1 & 2 & 0 \\ -1 & 5 & 2 \end{bmatrix}$ B=$\begin{bmatrix} 1 & 2 \\ -1 & 0 \\ 3 & 1 \end{bmatrix}$ i ...
3
votes
1answer
11 views

Linear independent set of functionals makes certain map surjective

Let $V$ be a finite $n$-dimensional vector space over a field $K$ and $\{\lambda_{1},\ldots, \lambda_{n}\}$ be a linearly independent set of functionals Show that the linear map $$\Lambda:V\to K^n$$ ...
1
vote
4answers
61 views

How Many Times The Clock Hands Make an angle Theta?

You might have encountered such a question where $\theta=90^o$ but this a little different and is causing a little problem. I approached this problem in the following matter and solved for $\theta= ...
0
votes
3answers
99 views

When $\operatorname{im}(A) = \ker(A)$

Consider the following true/ false qustion: There exists a $2 \times 2$ matrix $A$ such that $\operatorname{im}(A) = \ker(A)$. I know that this is true, but I am not sure how to show it. If $A$ ...
0
votes
0answers
18 views

Proof that $\operatorname{End}(V) \rightarrow Gl_n(K), F \mapsto M_A^A(F)$ denotes a group-isomoprhism.

Definition: Let $A$ be a Basis of $V$, $V$ a $K$ - Vectorspace. $M_A^A(F) = \Phi_A \circ F \circ \Phi_A^{-1} $, where $\Phi_A$ denotes the following function: $n := \dim V, \{x_1,\ldots,x_n\} = A$ ...
1
vote
1answer
25 views

Anticommuting matrices and their eigenvalues

Let $A,B\in \mathcal{M}_n(\mathbb{C})$. It is known that if $AB=BA$ and $\lambda_1, \lambda_2, \dots, \lambda_n $ are the eigenvalues of $A$ and $\beta_1, \beta_2, \dots, \beta_n$ are the ...
0
votes
0answers
11 views

Unitarily equivalent Triangular matrices

Could anyone help me to prove the following problem? Suppose $(x_1,x_2,\dots,x_n)$ is a permutation of $(y_1,y_2,\dots,y_n)$, then any triangular matrix with diagonal entries $(x_1,x_2,\dots,x_n)$ is ...
1
vote
0answers
41 views

Spliting subspaces and fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
0
votes
0answers
15 views

Determining if a Polynomial is a subspace and its Basis

Hi, the question is Which of the subsets of P2 given in Exercises 1 through 5 are subspaces of P2 Find a basis for those that are subspaces. So I know that P'(1) = 1b + 2c And I know that P(2) = a ...
-1
votes
3answers
38 views

A is a square matrix and given that $A^3 = 2\mathbb{I}$, then show $A-\mathbb{I}$ is invertible and find its inverse [on hold]

Could anyone guide along with this question? I was trying $(A−I)(A−I)^{−1} = I$ and was figuring if there's a way out to expand $(A−I)^{−1}$. I also tried $(A−I)x=0$ but to no avail.
1
vote
1answer
37 views

Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
0
votes
2answers
28 views

If null(AB) is a subset of null(A), does they have the same rank?

Let $A$ and $B$ be a square matrices. If every solution to $AB_x=0$ is also a solution to $A_x=0$ then $rank(AB)$ = $rank(A)$. I'm not sure if the logic is good here : $AB_x=0 \;\;and\;\;A_x=0\; ...
0
votes
3answers
42 views

Cross product and matrix of rotation

I am looking for simplify the following equation and extract vector $\omega$ to the right side. $(R\cdot x)\times(R\cdot(\omega\times x))$ where $\times$ is the three-dimensional cross product, $x$ ...
0
votes
2answers
24 views

Null space is an Invariant subspace

Let $\lambda$ be an eigenvalue of a square matrix $A$. Show the null space of $(A-\lambda I)^j$ is an A-Invariant subspace of $\mathbb{C}^n$ for all positive integers $j$. Proof without requiring ...
0
votes
3answers
63 views

If $1,-1,0$ are eigen values of $A$ then $\det(I+A^{100})=$?

As the question states, if $1,-1,0$ are eigen values of a matrix $A$ then I need to find what $\det(I+A^{100})$ is. Now I know that $\det A=0$, $\det (I+A)=0$ and $\det(I-A)=0$. But I don't know what ...
2
votes
0answers
20 views

dimension and base of arithmetic sequence

Arithmetic sequence is a vector space. But how to find a dimension of it. I try this: arithmetic sequence: $a_1,a_2,...,a_n={a_1,(a_1+d),(a_1+2d),...,(a_n+(n-1)d)}$ With only $a_1$ and $d=(a_1-a_2$) ...
1
vote
1answer
38 views

inequality with a positive matrix

Let $$ A=\left[ \begin{array}{cc} a & b\\ \overline{b} & c\\ \end{array} \right]$$ be a positive semi-definite positive of $M_2(\mathbb{C})$. How prove the inequality $ac \geq ...
0
votes
0answers
25 views

Find orthogonal projection to x-y, x-z, and z-y, plane

In linear transformation from $R^3$ to $R^3$, how would you find the matrix of the linear transformations to do these projections?
0
votes
1answer
16 views

eigenvalue and rank of a transformation

what i feel is that since the range of the linear transformation is strictly less than $n$ this implies that the transformation is not onto hence the null space contains a non trivial vector.but is ...
2
votes
1answer
24 views

Tensor Product of Spaces has Basis of Tensor Products

I am given the following definition of the Tensor Product of spaces Given two vector spaces $V,W$ a vector space S is a tensor product of $V,W$ if there exists a map $M$ $$ M: V \times W ...