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 ...

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Help me construct a bijection for $\left(1..n\right)^m$, with restrictions (interlacing components)

Let $x \in \left(0..n-1\right)^m$. I want to construct a bijection $g$ : $\left(0..n-1\right)^m \to \left(0..n-1\right)^m$ such that if we know $m'$ components of $g(x)$ and $m-m'$ components of $x$, ...
3
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1answer
26 views

Is this an existing matrix distance/metric?

I was thinking about comparing different basis transformations and came up with this distance function: $$d(A,B)= \dfrac{||A - B||^2}{||A|| + ||B||}$$ I am using the Schatten-1-norm as the norm here ...
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2answers
26 views

How do I find a relation for these polynomials from a matrix?

I have the following three polynomials: $1 + 2t^2, 4 + t + 5t^2, 3 + 2t$. I need to show that they are linearly dependent in $\mathbb P_2$ (polynomials of degree at most $2$). I put them in a $3x3$ ...
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0answers
33 views

reference required for a well-known result

Consider the following classic problem: Given a 2-dimensional cube (a square) $[0,1]^2$, one of the minimum (area) simplex covering the 2-cube is bounded by $x_1 =0 , x_2=0, x_1+x_2 =2$. For ...
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0answers
9 views

Prove that in cyclic codes, ($C_1$+$C_2$)$^\perp$=$C_1^\perp$+$C_2^\perp$

Let $C_1$ and $C_2$ be cyclic codes over finite field with the same length. Prove that ($C_1$+$C_2$)$^\perp$=$C_1^\perp$+$C_2^\perp$. The direct conclusion is clear but how to prove the reverse ...
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0answers
14 views

the examples of subspace embedding which are not Oblivious

For the definitions of Oblivious Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf. Then, can any one show the examples of subspace embedding which are ...
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1answer
20 views

Linear Algebra - “Closest” Solution

Suppose I have a combination of vectors (alpha and beta unrestricted): $ \gamma = \alpha\begin{pmatrix} 3 \\ 2 \\ 5 \\ \end{pmatrix} + \beta \begin{pmatrix} 6 \\ 1 \\ 2 \\ ...
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2answers
21 views

nonhomogeneous case Ax=b of a singular matrix A

Prove that the nonhomogeneous case Ax=b has no solution unless (b,y)=0, for all vectors y satisfying A* y = 0, where A* is the adjoint of A, and A is singular. I'm not sure how to start this. I know ...
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0answers
7 views

the difference and similarity between 'subspace embedding' and ' dimension reduction'

Can someone show me the difference and similarity between 'subspace embedding' and 'dimension reduction' using the mathematical definition?
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0answers
27 views

Squaring a matrix using a linear memory

I have a N x N matrix (let's denote it with A). I want to calculate $A ^ 2$, using $\theta(N)$ memory (speed does not matter as long as it's a polynomial) on one processor. I believe that this can ...
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4answers
24 views

Determine if a given set of vectors span $\mathbb{R}[x]_{\leq2}$

I am working on a linear algebra question, which asks you to determine if the vectors $(1+x), (1-x), x, (1+x^2)$ span the vector space $V=\mathbb{R}[x]_{\leq2}$. I think that the four vectors do ...
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0answers
19 views

Linear transformation of eigenspace is subset of eigenspace

Let $V$ be a vector space over a field $\mathbb{F}$ and let $L$, $M$ be two linear transformations from $V$ to itself. a. Show that the subset $W= {x ∈ V : L(x) = M(x)}$ is a subspace of $V$ b. ...
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0answers
19 views

Finding the largest singular value “easily”

Im only interested in finding the largest singular value. I don't need the singular vectors. Is there a way to do so without performing full SVD? Is there an analytical expression? If not, is ...
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0answers
36 views

using the principle of mathematical induction [on hold]

Can you tell me how to find the second equation using the first one with induction
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2answers
19 views

Find the matrix representation of linear transformation $T$

Suppose that $T(1, 1) = (3, 1, 0)$ and $T(0, 2) = (3, 0, 1)$, find the matrix representation of $T$. So the canonical bases for $\mathbb{R}^2$ are $(0, 1)$ and $(1, 0)$. Starting with those I have ...
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1answer
11 views

Proving distance inequality between three elements in a normed linear space

For any two elements $x,y$ belonging to a normed linear space, distance between x and y is given by $\rho(x,y) = ||x-y||$ I am trying to prove the inequality $\rho(x,y) \leq \rho(x,z) + \rho(y,z)$ ...
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0answers
12 views

Vector norm of $\mathbb R^n$, why is $p$-norm$\leq q$-norm if $p\geq q$? [on hold]

Considering $$\infty\geq p\geq q\geq1$$ How can I show that the $p$-norm is smaller or equal to the $q$-norm? I can only show the case for $p=1, q=2$, but have no idea how to show others. Thank ...
0
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1answer
17 views

generalized inverse in the theory of projective module

A module P over a ring R is projective which is an important topic in the theory of commutative ring due to its structural property of being a direct summand of free module. But my question is why ...
3
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1answer
25 views

Is a square matrix with positive determinant, positive diagonal entries and negative off-diagonal entries an M-matrix?

I'm trying to determine if a certain class of matrices are M-matrices in general. I'm considering square matrices $A$ with the following properties: $\det(A) > 0$ (strictly), all the diagonal ...
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1answer
27 views

Matrix of a transformation with respect to bases a and b

Let $T:R^2\to R^2$ be the linear transformation $T:R^2\to R^2$ given by $$T\begin{bmatrix} x_1\\ x_2 \end{bmatrix} = \begin{bmatrix} -3x_1-3x_2 \\ 6x_1-x_2 \end{bmatrix}$$ (Pretend that's a 2x1 ...
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0answers
19 views

Linear Algebra, Stephen Friedberg, chapter Determinants [on hold]

Let $V$ be the vector space $M_{n\times n}(F)$ and $B$ a fixed $n\times n$ matrix from $V$. Let also $L_B$ and $R_B$ (left, right multiplication) in $V$ defined by $L_B(A)=BA$ and $R_B(B)=AB$. ...
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1answer
16 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V..

Let us assume that every vector in S_2 is a linear combination of vectors in S_1. Question: Does that mean that S_1 and S_2 are bases for the same subspace of V? I know that the answer to this ...
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2answers
17 views

Inner Product spaces with functions?

I understand inner product space with vectors, but the conversion to functions is throwing me off. Also why do they use an integral here, I've always seen summations. I think I'm missing something ...
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1answer
17 views

Inner product space related to pythagorean theorem.

So I understand that the an inner product space basically uses pythagorean theorem because it is similar to a distance formula. I'm still having trouble with this proof. I am a bit confused about ...
2
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2answers
44 views

Prove that $T$ is one to one?

Let $\alpha>0$. Assume that $T:V\to V$ is a linear operator that has the property that $$ \Vert T(x)\Vert\geq\alpha\Vert x\Vert $$ Show that $T$ must be one-to-one. I'm very confused on how to do ...
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0answers
16 views

can any one tighten $|e_i^TXe_j|$?

Suppose we have the symmetric matrix $X\in R^{m\times m}$ with its 2-norm $\|X\|_2\leq m$. Then I can get that, for each entry of $X$, $|X_{ij}|=|e_i^TXe_j|\leq\|e_i\|_2\|X\|_2\|\|e_j\|_2=m$, where ...
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0answers
27 views

show linear transformation bijective

Can you please help me prove this? Let $T:\mathbb{R}^7\to\mathbb{R}^7$ be a linear transformation such that 9 is an eigenvalue of $T$ and $dim(E_9)=6$ Prove that either T-4I or T-5I is a bijection ...
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2answers
18 views

$R(T)$ and $N(T)$ are $T$-invariant subspaces

Let $T:V\to V$ be linear. Show that $R(T)$ and $N(T)$ are $T$-invariant. I know that $\dim(V)=\dim(N(T))+\dim(R(T))$, but I'm confused on where to go from here.
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2answers
40 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.

Let us assume that every vector in $S_2$ is a linear combination of vectors in $S_1$. Question: Does that mean that $S_1$ and $S_2$ are bases for the same subspace of $V$? I know that the answer to ...
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0answers
12 views

How to compute $proj_wu$; $u$ vector onto $W$ span

Let $u = (1,-2,1,6)$ in $R^4$, and let $W$ = span${(1,1,-1,0),(1,1,0,0)}$ . Compute $proj_wu$ . My Question: Since this is not an orthogonal basis, should I use the Gram-Schmidt process to convert ...
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0answers
24 views

Operations with Big Matrices [on hold]

I was able to bring to the scilab the matrices 43x43 using the read command. It Worked! But now i got this error when I try to do operations with the two matrices. -->M.*MD !--error 9999 ...
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3answers
54 views

Determine whether the set $\{v_1 + v_2 - v_3, 2v_1 + 2v_3, -v_1 + v_2 - 3v_3\}$ is linearly dependent or independent.

We had a question on our last test that was very similar to this and I only got $2$ points of $6$ and I want to make sure I do it right this time. Here's my solution to that one: Let $v_1, v_2,$ and ...
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4answers
28 views

Let V be a vector space and W a subset of V. Suppose zero is in W and W is closed under addition. Is W a subspace of V?

I know that the answer to this question is No. My question is why is the answer no? What's missing? if possible give a specific example of both V and W such that W satisfies above conditoins but it ...
0
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1answer
30 views

Matrix representation of a transformation

We have a linear transformation $T: M_{2\times 2}(F) \to F$ by $T(A) = tr(A)$. We want to compute the matrix representation $[T]$ from $\alpha$ to $\gamma$ coordinates. $M_{2\times 2}$ has the ...
3
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1answer
16 views

A question about biorthogonal basis composed of eigenvectors of a finite-dimensional non-self-adjoint matrix

The non-self-adjoint matrix M has non-degenerate eigenvalues, that is $M \psi_i = e_i \psi_i$, and its adjoint matrix satisfies $M^\dagger \chi_j= e_j^* \chi_j$. I know that $(\chi_j, \psi_i) = ...
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1answer
18 views

How do I find a basis for the following subspace?

I'm unsure how to do the following problem: Find a basis of the following subspace of $R^4$. W = all vectors of the form $(a,b,c,d)$ where $a+b-c+d=0$. Any help would be great, many thanks :)
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3answers
21 views

Proving kerT is a subspace of V. and rangeT is a subspace of W.

My question is as follows: Suppose $V$ and $W$ are vector spaces, and let $T: V \longrightarrow W$ be a linear transformation. Show that $\ker T$ is a subspace of $V$. Show that ...
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2answers
19 views

How do you prove that T and U are the same linear transformation on an inner product space V?

Is it enough to show that $<T(x),y>$ = $<U(x),y>$ for any x and y in V?
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1answer
17 views

Solving a system of equations containing complex numbers - Gaussian elimination

Problem: Determine the solutions in $\mathbb{C}^3$ of the following system over $\mathbb{C}$: \begin{align*} \begin{cases} 2x+iy-(1+i)z &=1 \\ x-2y+ iz &= 0 \\ -ix +y -(2-i)z &= 1 ...
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1answer
8 views

Determine rank and nullity of linear transformation between polynomial of degree $\leq$ 5 to $R^6$

Define the mapping $T$to be the one that maps a polynomial $f(x) \in V$ to the vector $(f(0), f(1),f(2),f(3),f(4),f(5))^t$, where $V$ is the vector space of all real polynomials of degree 5 or less. ...
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1answer
27 views

Not understanding what linear groups are, please need help on the questions 1-4

Above is my math homework. I am in a linear algebra class that is the first linear algebra course i am taken and am overwhelmed with the problem. I am not understanding what to do, but i understand ...
2
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2answers
31 views

Find the slope of the line that goes through the given points

I know the formula for this type of problem is the second y coordinate subtracted from the first y coordinate over the second x coordinate subtracted from the first x coordinate but for the numbers ...
0
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2answers
35 views

Trouble finding Jordan Normal form for $4 \times $ 4 matrix

$M = \left(\begin{array}{cccc}0 & 1 & 0 & 0 \\-3 & 4 & 0 & 0 \\2 & -1 & 2 & 0 \\-1 & 1 & 1 & 2\end{array}\right)$. I find the eigenvalues to be ...
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1answer
24 views

Is Basis of a vector space a subset of the vector space

Now, I was going through my notes which says that basis of a vector space V is a set S such that 1)S is a linearly independent set 2)v=L(S) Now there might be multiple basis of a vector space.Hence ...
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1answer
22 views

Writing a matrix in terms of a basis

I've looked for examples but found none similar to this; I have $\mathfrak{sl}(2,K)$ with the given basis $S$ as follows: $S=\{e,h,f\}$ where $e = \pmatrix{0 & 1 \\ 0 & 0}$ $h = \pmatrix{1 ...
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0answers
21 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
1
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1answer
19 views

Linear Algebra Orthogonality Help

I am struggling with this one exercise from self-learning. I simply do not understand what it is asking. If someone could walk me through this problem I would be very grateful.
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0answers
23 views

Expanding linear functional to base of $V^*$

Given a linear functional $f_1\in V^*$ where $V^*$ is a dual space of $V$, I can expand it to the base of $V^*$ : $B^*=\{f_1,f_2,...,f_n\}$, that I know. But does it mean that exist a base $B$ for $V$ ...
0
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0answers
33 views

Given $x\in\mathbb R^n$ and an $m\times n$ matrix $A$, show that $x\in \ker A$ or not. [on hold]

Given $x\in\mathbb R^n$ and an $m\times n$ matrix $A$, show that $x\in \ker A$ or not. I understand that the solution to $\ker A$ is the set of all solutions to $Ax=0$. I'm confused about how I ...
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votes
1answer
28 views

Lift-club rates (This should be really easy)

Right, this is actually a real-life problem. I want to join Bob and Joe's lift club. Joe usually pays about \$40 a week (in total) to drive between A and B (for fuel). (Driving from A to B and back is ...