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
1answer
113 views

A linear dependent set contained in a vector space

If a set of vector space V over the field F contains a linear dependent set S, then can we say that V is linearly dependent? This is my reasonning, it makes sense to me, I just don't know if it is ...
1
vote
3answers
2k views

Suppose that each of the row sums of an $n\times n$ matrix $A$ is equal to zero. Show that $A$ must be singular.

Suppose that each of the row sums of an $n\times n$ matrix $A$ is equal to zero. Show that $A$ must be singular. I can show this for $2\times2$ by using the fact that $det(A)=0$, whenever $A$ is ...
0
votes
1answer
47 views

Equivalence of singular matrix properties (2)

So another implication I can't manage to prove is the following: Let $A$ be a $2 \times 2$ matrix, I) $A\vec{v}=\vec{w}$ for some $\vec{v}\neq\vec{w}$ II) $A$ has no inverse. To show: $I ...
1
vote
0answers
63 views

Asymptotic behaviour of sum of inverse matrix elements

I was wondering if anyone knows if some theory exists on the following problem. I'm considering the minimization problem $h^2\boldsymbol{v}M_N\boldsymbol{v}$ subject to ...
1
vote
1answer
75 views

Which of these characteristic polynomials imply that the matrix can split?

This a paraphrase of another old exam problem: Let $M$ be a $\:5\times 5\:$ matrix with rational entries whose characteristic polynomial takes the form $\;\left(x^{\hspace{.02 in}2}+1\right) ...
0
votes
1answer
112 views

Equivalence of singular matrix properties

I need to prove the following: $A$ has no inverse $\Rightarrow\det(A)=0$. I know it is simple but I just do not know how... $A=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ has no inverse ...
0
votes
1answer
124 views

How do I find the probability of these three events?

Sample space is all houses in zip code 80210. Event A is that the houses were built in 2002, event B is that the houses have 3-car garages, and event C is that the houses have lead-base paint. ...
5
votes
4answers
809 views

Meaning of signed volume

I want to understand the definition of the determinant of a $n\times n$ real matrix $A$ as the signed volume of the image of the unit cube $C'$ under the linear transformation given by $A$, i.e. $x\to ...
0
votes
2answers
81 views

simple question: the eigenvectors of a matrix.

\begin{align*}A=\left(\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \\\end{array}\right);\end{align*} The ...
3
votes
1answer
150 views

Unitary operator on $\mathbb{R}^2$

I am trying to solve the following froblem. Any unitary operator on $\mathbb{R}^2$ in standard inner product with standard ordered basis is given by any of the following two matrices $$A = ...
0
votes
2answers
36 views

Find the axis of reflection

I have to determine the axis of reflection of the composition of a rotation and a reflection, y show that the order of composition matters. So I multiply the matrices that represent each isometry, ...
0
votes
3answers
144 views

Simple formula to get any linearly independent vector from any non-zero lenght vector in 3D.

In 2d I can get a linearly independent vector from any non-zero length vector $(x,y)$ by using $(-y,x)$ [because: $x*-y+y*x=0$] Is there any similar simple expression to obtain any linearly ...
1
vote
0answers
134 views

non trivial rotation invariant subspaces of $\Bbb R^2$ and $\Bbb C^2$

i have two questions. First one is: am i correct in thinking that a line vector subspace of $\Bbb R^2$ is rotation invariant if $\theta = \pi$ or $2\pi$? Second, i am told that there is a non ...
2
votes
2answers
2k views

What is the difference between Linear Transform and Linear Operator

Try to understand the difference between Linear transformation and Linear Operator? What we learn in linear algebra class for linear transformation: ...
0
votes
1answer
42 views

Row equivalency of matrices

Let $M$ be an $n\times m$ matrix and let $N$ be an invertible $n\times n$ matrix. Why is it that the matrix $NM$ is row equivalent to $M$? Thank you in advance.
2
votes
1answer
175 views

Task interpretation for Hoffman Kunze Linear Algebra exercise 1 (b) sec. 3.6

I don't understand the task from (b). Is it equivalent to : for every linear functional $f(x_{1}, ..., x_{n})=c_{1}x_{1}+...+c_{n}x_{n}$ on $F^{n}$ which satisfy $c_{1}+...+c_{n}=0$ there exists ...
3
votes
0answers
80 views

Set of commuting Matrices $\Rightarrow $ Common Eigenvector [duplicate]

I am trying to prove that if we have an arbitrary set of commuting matrices in $M_n(\mathbb C)$ then they have a common eigenvector. Well, if we have only 2 matrices, the answer is easy and it has ...
0
votes
4answers
11k views

Find a plane that passes through a point and is parallel to a given plane

Find an equation of a plane that passes through the point $(0, 1, 0)$ and is parallel to the plane $4x - 3y + 5z = 0$ I first plugged the "missing" variable $4x - 3y + 5z - d = 0$ then ...
4
votes
3answers
710 views

Column Space and SVD

I was reading Gilbert Strang's book and he says that if $A=USV'$ be the SVD of A ( assume square for the moment) then the nullspace of A is given by the last $n-r$ columns of V and the column space by ...
0
votes
1answer
534 views

Prove that a system is consistent

Prove the that system $\begin{cases} x + 2y - z = a\\ 2x +y + 3z = b\\ x - 4y +9z = c \end{cases}$ is consistent when $c = 2b - 3a$ I am assuming I need to show this with row operation and that the ...
0
votes
1answer
211 views

Two approaches to computing angle between two real vectors in higher dimensions.

Let $u,v$ be a pair of linearly independent vectors in $\mathbb{R}^n$. The angle between the two vectors $\theta$ is given by $\arccos \left( \dfrac{u \cdot v}{\|u\| \|v\|} \right)$ where the dot ...
2
votes
0answers
71 views

Orthogonal Latin squares - Origin of the word “Orthogonal”

Is there any linear-algebraic link with the use of the word "orthogonal" in orthogonal latin squares? I thought about it a little bit and the closest I got to linear algebra was this definition : if ...
0
votes
1answer
56 views

Projection self-adjoint

Let $A$ be a projection such that $A^2=A$ then I want to prove that $A=A^* \Leftrightarrow \ker(A) \perp \operatorname{im}(A)$. The implication $A=A^*\Rightarrow \ker(A) \perp ...
0
votes
0answers
182 views

Help on a coset proof

I want to see if I'm right, and if not, a possible solution. Problem Statement: In the additive group $R^m$ of vectors, let $W$ be the set of solutions of a system of homogeneous linear equation ...
1
vote
1answer
75 views

Iteration to find squre root of positive semidefinite matrix

Suppose matrix $A$ is positive semidefinite and $I\succeq A$. Prove that the iteration $$Y_0=0,\hspace{3mm} Y_{n+1}=\frac{1}{2}(A+Y_n^2)$$ is nondecreasing (that is, $Y_{n+1}\succeq Y_n$ for all ...
2
votes
1answer
168 views

Left multiplication by invertible matrix.

If M is an $n\times m$ matrix of rank $n$ and $A\in GL_n$, that is, $A$ is an invertible $n\times n$ matrix, then why is the row reduced echelon form of $M$ is the same as the row reduced echelon form ...
0
votes
1answer
62 views

What is $f^TAf$ intuitively

I can get from the definition of matrix production that the result of the formula is $\Sigma_{i=0}^nf_ia_{ij}f_j$. But is there any physical interpretation of this formula, like some sort of ...
1
vote
2answers
3k views

Composition of two reflections is a rotation

I have this problem that says: Prove that in the plane, every rotation about the origin is composition of two reflections in axis on the origin. First I have to say that this is a translation, off my ...
2
votes
2answers
249 views

Given factorization A = QR where Q's columns are pairwise orthogonal, but not orthonormal, how do i normalize Q's columns?

My questions is: Given a factorization A = QR where Q's columns are pairwise orthogonal, but not orthonormal, how do i normalize Q's columns while transforming R so result is still equal to A ? I ...
4
votes
1answer
5k views

What's a non-zero (column) vector?

This is a basic question, but I can't find the definition on wikipedia, google, or math.stackexchange, because I only find examples of it being used in problems. Therefore, I want to clarify: Does a ...
0
votes
2answers
63 views

a question about linear subspaces of $\mathbb{R}^n$

Say I have a $\mathbb{R}^n$ and $E,F$ are a $k$-dimensional subspaces of $\mathbb{R}^n$. If $E^{\perp}\cap F=\{0\}$, Does it follow that the projection of $F$ onto $E$ is all of $E$? (say we're using ...
1
vote
1answer
51 views

Diagonalization of order 2 matrix

Consider $A\in Gl(n,{\bf R})$ such that $A^2=I$. So we have eigenvalue $\lambda^2=1$ on ${\bf C}$. This implies $\lambda = \pm 1$. Here I want to show that $A$ is a diagonalization whose entries ...
2
votes
3answers
122 views

linear independence question

I'm asked to show linear independence for the following familiy of functions as functions on the interval $[-\pi, \pi]$: 1) $A= \{f_1,\ f_2, \ f_3 \}$ where $\ f_1(x)=1$, $\ f_2(x)=\cos x$, and $\ ...
2
votes
1answer
271 views

Theorem 20 Hoffman Kunze Linear Algebra book Section 3.6

What I don't understand here is why is $h(\alpha)=0$ for all $\alpha$ in $N_{k}$. Is there a typo? In case there is not, could someone please detail that last step please?
2
votes
1answer
130 views

The function $\langle \cdot , \cdot \rangle_A : \mathbb{R}^n \times \mathbb{R}^n\rightarrow \mathbb{R}: (v,w) \mapsto \langle Av, w\rangle$

here I've got a linear algebra related question: For which set of matrices contained in $\mathbb{M}^{n \times n} \ $is the function $\langle \cdot , \cdot \rangle_A : \mathbb{R}^n \times ...
1
vote
2answers
162 views

Change of basis (Gram-Schmidt)

I was wondering whether it is possible to write down explicitely the matrix that represents the change of basis from a basis $\{v_1,....,v_n\}$ to a basis $\{e_1,...,e_n\}$, where $e_i$ is the basis ...
0
votes
1answer
60 views

Linear dependence vectors proof

Let vectors $v_1,v_2,\dots, v_n$ be linearly independent. Let $v$ be a vector s.t. $v,v_1,v_2,\dots,v_n$ are linearly dependent. How do I prove that $v$ is a linear combination of ...
3
votes
0answers
81 views

Problem involving subspaces and linear transformations

I'm asking for some opinions about my proof! $V$ and $W$ are vector spaces, and $T : V \rightarrow W$ is a linear transformation. $Z$ is a subspace of $W$, and $U$ is the set of all $\textbf{x} \in ...
3
votes
1answer
85 views

Finding all alternating bilinear $T$ that preserve a certain group of isometries of $\mathbb{R}^{n+1}$

Let $$G=\left\{\begin{pmatrix} H & 0 \\ 0 & 1\end{pmatrix} \ | \ H\in O(n), HJ=JH \right\}\subset \mathrm{Lin}(\mathbb{R}^{n+1},\mathbb{R}^{n+1}) $$ where: $n=2m$, $J$ is the standard complex ...
1
vote
1answer
322 views

Intermediate textbook in Linear Algebra

I am looking for a Linear Algebra textbook that for those who just finished elementary Linear Algebra. I just finished Introductory to Linear Algebra by Strang and read to Least Squares and ...
4
votes
3answers
145 views

Calculating determinant with real number on diagonal and units everywhere else

I'm solving a problem and I'm having difficulties in calculation of the determinants of two matrices. There is two $N\times N$ matrices: $$\left( \begin{array}{cccc} a & 1 & \ldots ...
6
votes
2answers
142 views

$v_1,v_2$ are eigenvectors of $A$. Is it true that $v_1-v_2$ is eigenvector of $A$?

Let $A \in \mathcal{M}_{12 \times 12}$ and let $v_1,v_2$ be eigenvectors of $A$ such that $Av_1 = v_1$ and $Av_2 = 2v_2$. Is it true that vector $v_1 - v_2$ is not eigenvector of $A$? My answer: We ...
8
votes
1answer
3k views

Direct Sum of vector subspaces

How is direct sum of two vector subspaces different from the sum of two vector subspaces i.e. how is $X\oplus Y$ different from $X + Y$, where $X, Y$ are subspaces.
0
votes
2answers
166 views

Vector Subspace question

$X,Y,Z$ are subspaces of vector space $V$ s.t. $X + Z = Y + Z$, then is $X= Y.$ How to prove it? If $X \oplus Z= Y\oplus Z$ then is $X = Y.$
4
votes
1answer
91 views

Expressing T in terms of its adjoint for T normal

The question is to show that if $T$ is normal, there exists a unitary operator $U$ such that $T^{*}=UT$. My guess is that we use the polar decomposition of $T$- into a product of a unitary and ...
4
votes
2answers
189 views

Prove that the set of all diagonal matrices is a subring of $\operatorname{Mat}_n(R)$ which is isomorphic to $R \times\dots\times R$ ($n$ factors)

Can someone tell me, is that diagonal matrices is a subring of $\operatorname{Mat}_n(R)$ which is (ring) isomorphic to $R \times · · · \times R$ (n factors) and why?.
7
votes
4answers
168 views

characterize all matrices $X$ such that $BA = X$ whenever $AB = X$

It is clear that if $A$ and $B$ are $n\times n$ matrices (over a field) with $AB = I$ then $BA = I$. I like to characterize all matrices $X$ such that $BA = X$ whenever $AB = X$.
0
votes
2answers
115 views

Quotient ring of $T/I$

Please help me to identify the Quotient Ring of $T/I$, since $T$ is set of all triangular matrices, and $I$ is set of all strictly triangular matrices and $I$ is ideal in $T$. For your help I am ...
0
votes
1answer
32 views

Find the value of the variable in the combinatorics.

We are given the following inequality: $$\binom{n}{n/2}>T\;,$$ where $T$ is some fixed value. How can we find the value of $n$? Do we need to do hit and trial to get the solution or is there any ...
1
vote
2answers
47 views

Is linear space totally different from group?

Linear space builds on Abelian group. My question is, is linear space TOTALLY different from group? Is it true that some properties of linear space are the properties of the Abelian group? Actually, ...