Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, Hamel basis, dimension, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, etc. For questions specifically concerning ...
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1answer
53 views
$A^3 + A = 0$ then $rank (A) = 2$ [duplicate]
Let $A$ be a $3\times 3$ non-zero real matrix and satisfies $A^3 + A = 0$. Then prove that $rank (A) = 2$.
As $A$ is satisfying $A^3 + A = 0$, so $0$ is an eigen value of $A$.So $rank (A) < 3$. So ...
-3
votes
0answers
37 views
$A^ {12} = I$ hold for $A \in M_n(\mathbb Z)$ [closed]
Let us suppose $A \in M_n(\mathbb Z)$ and $A^n= I$ for some positive integer $n$ then prove that $A^ {12} = I$.
1
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0answers
23 views
Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?
Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow,
...
1
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0answers
34 views
I'm having trouble finding this matrix $T$ relative to $\mathcal B$ and the standard basis $\mathcal E$ for $\mathbb R^2$
This was a homework assignment, but unfortunately it was the last homework assignment of the semester so I never got feedback and I'm just reviewing it for a final. I'm supposed to let $\mathcal ...
0
votes
1answer
20 views
Relationship between 2 Dimensional Quadratic systems and roots
Given four points
$(x_1, y_1)
(x_2, y_2)
(x_3, y_3)
(x_4, y_4)$
How does one construct a system of two equations:
$a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$
$b_1x + b_2x^2 + b_3y + b_4y^2 + ...
0
votes
1answer
24 views
Finding the x value after a matrix multiplication?
I have the following solution of a problem, and I was wondering about a hopefully quite simple thing in it:
I was wondering how do they get from [5,10,5] to 5x? I am pretty sure there is a simple ...
7
votes
1answer
37 views
Show that if some nontrivial linear combination of vectors $\vec{u}$ and $\vec{v}$ is $\vec{0}$, then $\vec{u}$ and $\vec{v}$ are parallel.
I've never been that great at writing proofs, but I'm getting a bit better. I think I have the answer correct, but I don't know if I'm missing anything. My logic seems right but there may be some ...
1
vote
0answers
62 views
$\mathbb{R}^n$ and $\mathbb{Q}^n$: On the Nature of Solutions
I would just like to ask a simple question about solutions of non-homogenous linear equations both in $\mathbb{R}^n$ and $\mathbb{Q}^n$:
What does it mean that a system of non-homogenous linear ...
2
votes
1answer
35 views
Find the inverse for arbitrary k
I need to find a, b, c, d, e, f, g, h (all of which are not zero)
such that for all k is in Real number, show A is invertible or this can't happen
$$A = \left(\begin{array}{ccc}
...
4
votes
2answers
58 views
Positive semidefiniteness of a block matrix of positive semidefinite matrices
Given any symmetric matrix $\mathbf{M} = \begin{pmatrix}
\mathbf{A} & \mathbf{B}\\
\mathbf{B}^\mathrm{T}& \mathbf{C}
\end{pmatrix}$, the following conditions are equivalent:
(1) ...
0
votes
1answer
34 views
Proof $||A \underline x|| > 0 \Leftrightarrow \underline x \neq \underline 0$
If $n \geq m, A \in M(n,m)$ and $rg(A)=m$
Proof $||A \underline x|| > 0 \Leftrightarrow \underline x \neq \underline 0$
a)If $m \neq 0 \leftrightarrow A \neq 0_M$
Suppose that $\underline x = ...
1
vote
1answer
38 views
Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?
Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define $ \mathbf{x}'=(|x_1|,...,|x_n|) $ .
Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$ for every norm on $ \mathbb{R}^n $ ?
NB: The answer ...
0
votes
1answer
33 views
How to expand equation inside the L2-norm?
I want expand an L2-norm with some matrix operation inside.
Assume I have a regression $Y=X\beta+\epsilon$.
I want to solve (meaning expand),
$$\displaystyle\|Y-X\beta \|_{2}^2$$
Should I do:
1)
...
0
votes
0answers
30 views
Inverse of element wise power of a matrix
Is there any identity related to "inverse of elementwise power of a matrix".
In other words, it is well known that $[A^{n}]^{-1} = [A^{-1}]^n$, can we say something about $[A^{.n}]^{-1}$ where ...
0
votes
2answers
24 views
Matrix Norm Inequality
So I'm trying to prove that
$\lVert A\rVert_\infty \leq \sqrt{n} \lVert A\rVert_2$.
I've written the right hand side in terms of rows, but this method doesn't seem to be getting me anywhere.
Where ...
1
vote
2answers
33 views
Showing that $EF = 0$ and $EF^t = I_4$ without using determinants but images and kernels
We have two $4\times 4$ matrices, $E$ and $F$. And $EF = 0$, and $EF^t = I_4$, where $F^t$ denotes the transpose of $t$ and $I_4$ the identity matrix in $M_4(\mathbb{R})$.
By using determinants, ...
0
votes
3answers
65 views
$\xi$ be a primitive cube root of unity
Could anyone tell me first of all the below problem is wrong or okay?I am not able to figure out how to solve
$\xi$ be a primitive cube root of unity define ...
2
votes
1answer
61 views
Solving for a matrix from its quadratic form
I have a set of vectors that I am trying to predict from another set of vectors using a matrix $W$. To find this matrix, I decide I want to minimize the $\ell^2$ norm of the error, e.g.:
$$
...
2
votes
1answer
34 views
Vector 2 norm and infinity norm proof
So I've already proven why $\left\lVert x\right\rVert_2\geq \left\lVert x\right\rVert_\infty$. I'm having trouble proving that $\sqrt{m}{\left\lVert x\right\rVert_\infty}\geq \left\lVert ...
0
votes
0answers
23 views
Is this generally a necessary condition for Orthogonality of a finite set?
Let $v_1,...,v_n$ be indexed vectors in an inner product space $V$ such that $i\neq j\Rightarrow \langle v_i,v_j\rangle = 0$
In some texts, the condition "$i\neq j \Rightarrow v_i\neq v_j$" is ...
2
votes
2answers
74 views
Why do we need 2 equations to solve 2 variables, 3 equations to solve 3 variables, etc…?
The question put another way: why does solving a system of two equations with 2 variables give an exact answer?
I understand that using graphing, I will get either 2 parallel lines, which means 'no ...
-1
votes
0answers
17 views
Show that $\dim(W)=k$ where $W$ is $T-cyclic$ subspace of $V$ generated by $v=v_1+\dots+v_k$.
Show that $\dim(W)=k$ where $W$ is $T-cyclic$ subspace of $V$ generated by $v=v_1+\dots+v_k$.
Hmm.. It is true that $W$ has a basis {$v,T(v),\dots,T^{k-1}(v)$}
but I'm not sure how to use ...
1
vote
1answer
22 views
Eigenvalues of $\sum_{i=1}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}$
Consider the cuadratic form
$$
\mathbf{x}^{\intercal}Q\mathbf{x} = \frac{x_1^2}{\lambda_1} + \sum_{i=2}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}\ .
$$
Is it true that the eigenvalues of $Q$ are ...
-1
votes
0answers
29 views
matrix norm that is invariant through a change of orthogonal basis
I am looking for (real) matrix norms that would be invariant through a change of orthogonal basis. I know only about the Frobenius norm, which satisfy this condition as it can be expressed as a ...
2
votes
2answers
43 views
Finding diagonalizable matrix's basis
If $T: R^3 \rightarrow R^3$ is diagonalizable and has distinct eigenvalues, find $0 \neq v \in R^3$ such that
{$v,T(v), T^2(v)$} is a basis for $R^3$.
I think that statement: {$v,T(v), T^2(v)$} is ...
1
vote
2answers
22 views
Prove using an example that there is no plane on R3 that contains every group of 4 points
Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere):
The exercise is as follows:
i) Find the equation of the plane of R3 that ...
4
votes
3answers
70 views
How to show $T$ is diagonalizable?
Let $T\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be linear with distinct eigenvalues $\lambda_1, \lambda_2, \lambda_3$. Show that $T$ is diagonalizable.
It seems as if this is a very simple ...
0
votes
2answers
37 views
Finding Vectors in cartesian form
I am stuck on this question could you please help me.
Find,in Cartesian form, the equations of the straight line through the point with position vector (-1,2,-3) parallel to the direction given by ...
4
votes
2answers
49 views
Let $\alpha$ be a solution of a monic quadratic polynomial with integer entries and $|\alpha|=1$.Then prove that $\alpha^{12}=1$.
($a$) Let $\alpha$ be a solution of a monic quadratic polynomial with integer entries and $|\alpha|=1$.Then prove that $\alpha^{12}=1$.
($b$)Let $A \in M_2(\mathbb{Z})$ such that $A^n=I$ for some $n$ ...
0
votes
2answers
50 views
Common eigenvector of two linear transformation matrices
I have two linear transformation matrices
\begin{pmatrix}
3 & 2 \\
-2 & 1
\end{pmatrix}
and
\begin{pmatrix}
1-a & -a \\
a & 1
\end{pmatrix}
How to find out what the value of ...
0
votes
1answer
18 views
Quadratic form positive semidefinite if limits in every direction are nonnegative?
Let
$$q(x_1,\ldots,x_n) = \sum_{i,j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in \mathbb{R}.$$
be a quadratic form with real coefficients.
Suppose that the limit is nonnegative in every direction. That ...
1
vote
0answers
27 views
Decomposition of vectorspace and subspaces invariant under a linear operator
Are the following claims true or false?
(1) Let $T$ be a linear operator on a finite dimensional vector space $V$ and let $V=W_1 \bigoplus W_2$ where $W_1$ and $W_2$ are $T$-invariant subspaces of ...
1
vote
1answer
23 views
Quadratic form positive definite if limits are positive infinity?
Let
$$q(x_1,\ldots,x_n) = \sum_{i,j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in \mathbb{R}.$$
be a quadratic form with real coefficients. Suppose that
$$ \lim_{x_i \rightarrow \pm \infty} ...
2
votes
0answers
51 views
Real projective space is Hausdorff
I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix??
This prove is correct or I need to add something ?? ...
2
votes
2answers
25 views
Let $U$ and $V$ be two $n\times n$ unitary matrices such that $UV=\lambda VU$.Then prove that $\lambda^n=1$
Let $U$ and $V$ be two $n\times n$ unitary matrices such that $UV=\lambda VU$.Then prove that $\lambda^n=1$
Totally stuck on it.Can I get some help?
1
vote
1answer
40 views
generalized eigenvector for 3x3 matrix with 1 eigenvalue, 2 eigenvectors
I am trying to find a generalized eigenvector in this problem. (I understand the general theory goes much deeper, but we are only responsible for a limited number of cases.)
I have found ...
10
votes
0answers
80 views
Combinatorics in finite vector space
Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$.
Let $k$, $a$ and $b$ be non-negative integers.
Determine the number of subspaces $K$ of $V$ ...
2
votes
2answers
86 views
If $A$ is invertible show that $\det(A) \neq 0$
If $A \in M_{n\times n}(F)$ is not invertible then the rank of $A$ is less than $n$,
thus $\det(A) =0$.
I proved that way, but looks like too simple so I think maybe there is a trick that I missed.
...
2
votes
2answers
38 views
GHK = 0, What is the relationship between G and K if H is invertible?
I'm practicing a problem that I came across, it's
$\text{G}$, $\text{H}$, and $\text{K}$ are $4\times4$ matrices. $\text{GHK}=0$ and $\text{H}$ is invertible, does $\text{GK}=0$?
I was trying to ...
2
votes
3answers
54 views
What are some relationships between a matrix and its transpose?
All I can think of are that
If symmetric, they're equivalent
If A is orthogonal, then its transpose is equivalent to its inverse.
They have the same rank and determinant.
Is there any relationship ...
1
vote
1answer
34 views
dimension of a quotient space
$V$ is a vector space of polynomials of degree less than or equal to 50 and $W$ is the set of polynomials in $V$ which are divisible by $x^4$.
I have shown that $W$ is a subspace of $V$ and that the ...
3
votes
1answer
71 views
$AX=C$: An Inconsistent Linear Equation [duplicate]
Question:
Let $A \in M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$
has more than one solution. Prove that there is a column $C \in F^n$ such that
the system of linear ...
3
votes
1answer
59 views
Taking derivative of $L_0$-norm, $L_1$-norm, $L_2$-norm
I am a little confused about taking derivatives w.r.t. the norms.
$L_0$-norm: $L_0$ means number of non-zero elements in a vector. Say, I am interested in an $x_i$.
...
0
votes
2answers
44 views
Linearly independent subset?
If $u,v,w$ and $z$ are distinct elements in $R$, then
{$(1,u,u^2,u^3), (1,v,v^2,v^3), (1,w,w^2,w^3), (1,z,z^2,z^3)$} is a linearly independent subset of $R^4$.
Is that true or false?
I even can't ...
3
votes
1answer
41 views
Proving existence of unique polynomial satisfying some integral
I came across a problem that I was having some trouble with.
Fix a positive integer $n$. Let $f(x) \in C(R)$ be a (real valued) continuous function.
Show that there exists a unique polynomial $q(x) ...
1
vote
0answers
55 views
looking for problems and readings on subspaces and linear transformations for students
Math people:
I am teaching linear algebra. I like to teach subspaces and linear transformations early in the course. It justifies why one uses matrices and makes many properties of matrices, matrix ...
2
votes
0answers
21 views
Extending transvections/generating the symplectic group
The context is showing that the symplectic group is generated by symplectic transvections.
At the very bottom of http://www-math.mit.edu/~dav/sympgen.pdf it is stated that any transvection on the ...
1
vote
0answers
25 views
Question between symmetric matrix and transforming to other bases?
If $A$ is symmetric and $A^{10} = 0$, then $A$ must be $0$.
I was thinking that $A$ must have as many eigenvalues as it does rank, and from that statement,
one of its eigenvalues must be $0$, but ...
2
votes
0answers
32 views
Identities with Adjoints
The classical adjoint adjA of a square matrix A has its ij-th entry equal to the ji-th cofactor (signed minor) of A. If detA is not 0 we can define the inverse A* of A as (adjA)/detA and use ...
3
votes
2answers
256 views
+50
REVISITED$^2$: Solution in $\mathbb{R}^n \overset{?}{\implies}$ Solution in $\mathbb{Q}^n$
Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose that the system of linear
equations $AX = B$ has a solution in $\mathbb{R}^n$. Does it necessarily have a solution
in ...
