Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

learn more… | top users | synonyms

0
votes
2answers
31 views

Show that a unique matrix exists for the coordinate vectors in a vector space

If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space $V$, there exists a unique matrix $M$ such that for any $f\in V$, $[f]_A=M[f]_B$. My textbook uses this theorem ...
0
votes
0answers
16 views

Find the eigenvector for an operator on a linear span

Let $V$ be the linear span of the functions $1,cos(x),sin(x)$. Let the operator $T$ on $V$ be given by the rule $Ty(x)=y(x+ \pi/4)$. Find the eigenvalues and eigenvectors of T in V. I know how to ...
0
votes
1answer
40 views

Proofs for $n$-dimensional vector spaces $V$

Suppose $V$ is an $n$-dimensional vector space. Prove that there is at most $n$ linearly independent elements in $V$. Prove that a set of $m<n$ element in $V$ cannot span $V$. I'm not really ...
0
votes
1answer
33 views

Matrix raised to a power

Find $A^n$ for $n = 1,2,...$. Does $A^n$ tend to a limit? $$A= \begin{pmatrix} 4/5 & 2/5 \\ 1/5 & 3/5 \end{pmatrix}$$ I found the eigenvalues $\lambda=1,2/5$ and the eigenvectors ...
2
votes
1answer
16 views

One Note about One to one and Surjective of linear functional [on hold]

I read a note that: if $ f \neq 0$ is a linear functional on H, then f is onto (surjective) and it is not one to one (injective) in general. Why this is true? i think it need advance ...
0
votes
2answers
11 views

Find basis of 4x3 Matrix

I've been confused with the following matrix. I'm trying to find the basis of the image of A: $$A= \left( \begin{array}{ccc} 6&4&10\\ 4&-1&3\\ -2&-5&-7\\ -10&-3&-13 ...
1
vote
0answers
27 views

Proving that a set of functions is a vector space

We've given that $V$ is a vector space and that $L(V)$ the set with functions $T:V\rightarrow \mathbb{R}$ s.t. $T(a_1f_1+a_2f_2)=a_1T(f_1)+a_2T(f_2)$. We must show that $L(V)$ is a vector space. I ...
0
votes
0answers
35 views

Isomorphism of vector spaces

Let $S$ be the space of all $3\times k$ matrices,$T$ be the space of all column vectors consists of seven components.If $S$ is isomorphic to a subspace of $T$ then what are possible values of $k$? I ...
0
votes
0answers
13 views

Difference between homogenous and nonhomogenous linear systems?

What is the difference between these two? My book doesn't give any explanation at all? I don't understand why a homogenous equation Ax=0 has a nontrivial solution iff the equation has one free ...
-3
votes
0answers
13 views

How can y be used to factor N [duplicate]

Let x be the given muliple of φ(N). Then for any g in Z*[sub N] we have g^x=1 in Z[sub N]. Alice chooses a random g in Z∗[sub N] and computes the sequence ...
0
votes
1answer
24 views

How many ordered bases can be found for $\mathbb{Z}_p^n$ over filed $\mathbb{Z}_p$?

Take $\mathbb{Z}_p^n$ as a linear space over $\mathbb{Z}_p$. Now you can imagine multy bases for this space. (please leave a comment or have an edit if question is not clear enough.)
0
votes
1answer
21 views

Find a basis for the subspace of polynomials of degree 3

Let $\mathbb{P}_{3}$ be the collection of all polynomials of degree at most 3. Find a basis for the subspace consisting of those polynomials $p$ such that $p(1)=0$.
1
vote
2answers
57 views

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true?

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true? $1.$ $A=I$ $2.$ $\det(A)=1$ $3.$ $\operatorname{trace}(A)=n$ I have counter example for the first option.For ...
0
votes
1answer
16 views

Invertibe matrix is a transition matrix?

It is true that all transition matrices are invertible, but does the converse hold: All invertible matrices are transition matrices? I'm asking with regard to matrices over a field, but more general ...
0
votes
0answers
17 views

Division by 0 when solving linear equation using FFT with block circulant matrix

For a problem of Ax=b, if A is block circulant, let "a" be the first row of A, the problem is the same as circularconv(a, x)=b, therefore, x=ifft(fft(b)./fft(a)). However if I try a toy example with ...
0
votes
0answers
15 views

Use the Kronecker delta matrix to answer question

So I have the Kronecker delta which is denoted as $\delta_{ij}$=$I$. Let $b_1, b_2, \cdots, b_n$ be a set of $n$ real numbers, I must show that: $\sum\limits_{i=1}^n b_i \delta_{ij} = b_j$ and ...
0
votes
0answers
24 views

Is there any simple way of finding a matrix which commutes with a given (say, more complicated) matrix?

Suppose I want to find the eigenvectors and eigenvalues of a hermitian matrix $A$, but $A$ is big and ugly. Is there an easy way to find another, nicer, hermitian matrix $B$, such that $AB=BA$ and so ...
1
vote
1answer
80 views

True or False: Basis in the space of polynomials of degree less or equal to 2014 should contain polynomial of degree 2013. [on hold]

Basis in the space of polynomials of degree less or equal to 2014 should contain polynomial of degree 2013.
1
vote
1answer
19 views

Linear Spanning Functions [on hold]

Let $V$ be the linear span of the functions $1$, $\cos(x)$, $\cos(2x)$, $\cos(3x)$, and $\cos(4x)$. Is the function $\sin(x)$ in $V$? Justify your answer.
4
votes
1answer
40 views

Basis of the matrices with only non diagonalizable matrices

Is it possible to find a basis of $M_n(\mathbb{R})$ that only has non diagonalisable matrices ? I'm looking for a rather easy example, or a proof of the (non-)existence.
0
votes
0answers
25 views

What does it mean when a matrix row reduces to the identity matrix?

If a matrix row reduces to the identity matrix, what does that mean? The kernel is 0 vector? or basis for kernel is {}? Anything else?
1
vote
1answer
22 views

Proving that Mx is an eigenvector of B with λ as the eigenvalue given that $B=MAM^{-1}$ and $Ax=λx$

The square matrix A has λ as an eigenvalue with corresponding eigenvector x. The non-singular matrix M is of the same order as A. Show that Mx is an eigenvector of the matrix B, where $B = ...
2
votes
0answers
42 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
0
votes
1answer
26 views

Why does Givens rotation avoid iteration and Jacobi rotation doesn't in case of reducing a symmetric matrix to tridiagonal?

I am currently implementing symmetric matrix reduction to tridiagonal. I read that Givens rotation avoids iteration when it is used for reducing a matrix to tridiagonal whereas Jacobi rotation is ...
0
votes
0answers
30 views

statements of matrix analysis

Let $y$ be fixed value. Let $A=a(x,y)$ be a matrix and $f_{t}(x)=\frac{\sum_{n=0}^{\infty}{a^{(n)}(x,y)(\frac{1}{t})^n}}{\sum_{n=0}^{\infty}a^{(n)}(y,y)(\frac{1}{t})^n}$ Show that ...
0
votes
2answers
33 views

Can Cayley-Menger Determinant Be Negative?

Cayley-Menger determinant is used to calculate the area of a triangle, volume of a tetrahedron etc. Can be seen here. My question is; If given only positive numbers, can Cayley-Menger determinant ...
0
votes
0answers
18 views

If $\limsup_{t\to \infty} \int_{0}^{t}Tr(A(s))ds = \infty$ then $\limsup_{t\to \infty} |x(t)|=\infty$

For a homogeneous linear system of differential equations: $x'=Ax$ : Suppose that $\limsup_{t\to \infty} \int_{0}^{t}tr(A(s))ds = \infty$ ($tr(A):=$ trace of the matrix A). Then there exists solution ...
0
votes
1answer
21 views

subsapce of f(T)V T -invariant

On a vector space $X$, choose a nonzero element $v \in X$ and a linear map $T : V \to V$. $f(T)v$ is the space generated by $v, T(v), T^2(v),\dots$ I think any subspace of $f(T)v$ is also ...
0
votes
0answers
30 views

Determine matrix from linear transformation

Let $T_{1}$ and $T_{2}$ be linear transformations given by $$T_{1}([x_{1}, x_{2}])=[3x_{1}+5x_{2}, 4x_{1}+7x_{2}]$$ $$T_{2}([x_{1}, x_{2}])=[2x_{1}+9x_{2}, x_{1}+5x_{2}]$$ Find a matrix A such that ...
4
votes
1answer
36 views

Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
1
vote
1answer
22 views

How to determine the Jordan form and give a Jordan base for a matrix?

given is $\begin{pmatrix} 3&0&-1&0&0 \\ 1&3&0&1&0 \\ 0&0&3&0&0 \\ 0&0&0&3&0 \\ 0&0&0&0&-3 \end{pmatrix}$ I have to ...
0
votes
1answer
18 views

Skew symmetric Matrix - Commutative property

If A and B are two odd size skew symmetric matrices(for example $3 \times 3 $). Let us say $X=AB,Y=BA$ Question What is the general relationship between X and Y? Can we write Y using X?
0
votes
1answer
47 views

Different Definitions of Tensor product, Halmos, Formal Sums, Universal Property

In the classic Finite-Dimensional Vector Spaces by P. Halmos he defines the Tensor product as The tensor product $U \otimes V$ of two finite-dimensional vector spaces $U$ and $V$ (over the same ...
1
vote
1answer
70 views

Compute a 4 4 matrix M such that MA is the row-reduced echelon form of A.

Compute a 4 X 4 matrix M such that MA is the row-reduced echelon form of A. (Hint: M can be written as a product of elementary matrices.) A:= ...
-1
votes
1answer
16 views

Linear map problem

Given that $T$ is a linear map $T:\Bbb R^3 \to \Bbb R^2$ and that $$ T\left(\matrix{1\\0\\0}\right) = \left(\matrix{1\\0}\right)\qquad T\left(\matrix{1\\1\\0}\right) = \left(\matrix{0\\-3}\right) ...
1
vote
1answer
64 views

How to find maximum of an inverse of a matrix?

If there is a square $~n\times n~$ matrix $~H~$ where ALL the elements of $~H_{i,j}~$ are variables between two bounds, $~H_{i,j})_{min}~$ and $~H_{i,j})_{max}~$. Is there any relation to maximize ...
2
votes
0answers
38 views

Regarding Linear Subspaces over a Finite Field… TFAE:

Let $V=\mathbb{F}^n$, for a finite field $\mathbb{F}$. Prove the equivalence of the following statements: There is a linear subspace $C$ of $V$ with the property that every vector $v$ of ...
0
votes
1answer
15 views

Determine amount invested in each business [on hold]

An investor invested sh.$10000$ in two businesses A and B. He got returns of $15\%$ and $8\%$ at the end of the first economic year which amounted to sh. $4500$. Can you determine the amount he ...
0
votes
0answers
24 views

Why is QR algorithm using plane rotation followed by givens rotation better than just plane rotations?

To find eigenvectors from a tridiagonal matrix, it[ref:Numerical Recipes] says that QR algorithm using plane rotation followed by givens rotation(QR algorithm with implicit shifts) better than just ...
2
votes
0answers
22 views

If two vectors are orthogonal, linearly independent? [on hold]

If two vectors are orthogonal, two ventors are linearly independent? how can we show that?
0
votes
1answer
41 views

Which of the subsets of $\mathbb{R^{3\times 3}}$ are subspaces of $\mathbb{R^{3\times 3}}$?

The invertible $3 \times 3$ matrices The $3\times 3$ matrices whose entries are all integers The $3\times 3$ matrices with all zeros in the third row The non-invertible $3\times 3$ matrices The ...
0
votes
2answers
36 views

Equation of a curved line that passes through 3 points?

I have a screen wherein the upper-leftmost part is at x,y coordinate (0,0). Then I have a curved line that passes through 3 points: (132, 201), (295, 661) and (644, 1085). Now, say I want to find 7 ...
1
vote
1answer
37 views

Solving $Ax=B$: what's wrong with this linear algebra argument?

With $K>L$ and assuming that we are working with real variables, suppose that $B$ is $K\times 1$ and $A$ is $K\times L$ with full column rank. I'm trying to find $x$ ($L\times 1$) satisfying: $$ ...
0
votes
1answer
16 views

Standard basis for Complex vector space

What will be the standard basis of $\mathbb{C}^3$ or in general how can I find the standard basis for $\mathbb{C}^n$ ? Note: $\mathbb{C}$ is complex vector space
2
votes
0answers
42 views

Prove that solutions to linear system form a vector space of dimension $\geq 2$

I accept & appreciate any form of help with the following problem: $B_{nxn}$ "periodic matrix" with period $T$ such that $B(t+T) = B(t)$ for all $t\in \mathbb{R}$. Assume that the system $x' = ...
1
vote
0answers
24 views

Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
0
votes
0answers
14 views

Maximization over linear surjective mapping of polyhedron

I am reading this paper and confused about the derivation of equation (11) (page 3, bottom of column 2). I will rephrase it in this question. Let $\mathcal{P}_r = \{ x \in \mathbb{R}^n : P_r x \leq ...
0
votes
2answers
29 views

Help me understand my linear algebra book

If $(v_1,...,v_m)$ is a list of vectors in $V$, then each $v_j$ is a linear com- bination of $(v_1,...,v_m)$ (to show this, set $a_j = 1$ and let the other a’s in $a_1v_1 +···+a_mv_m$ equal 0.Thus ...
0
votes
2answers
33 views

Sign pattern symmetric matrices

I am interested in sign pattern symmetric real matrices ($a_{ij} a_{ji} \ge 0$ for all $i \ne j$). I have seen a published proof that such sign-symmetric matrices cannot have purely imaginary ...
1
vote
1answer
17 views

Basis and dimension of the null space and range

The linear map $T : \Bbb R^{n\times n}\to \Bbb R^{n\times n}$ is defined by the formula $$T(A) = \frac12(A+A^T)\;.$$ How do I find a basis of the null space of $T$ and determine its dimension? and ...