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

-1
votes
1answer
20 views

How do i solve this equation?

I need help with one of the equations that I'm going to have on my test: $$5-2x-\frac{5-3x}{2}=1$$
1
vote
0answers
12 views

How to find the basis of a matrix by using Gauss-Elliminaton?

I confuse that, This is my calculating process, Where i do the mistake in this process? I hope to understand this error.
0
votes
2answers
23 views

Linear system $AX=0$ has a nonzero solution

Which parameters $a,b,c,d$ satisfy the matrix $$A=\begin{bmatrix}-1 & 1&1&1 \\1 & -1&1&1\\1 &1 &-1 &1\\a&b&c&d \end{bmatrix}$$ sucht that linear system ...
0
votes
1answer
12 views

Find $x$ for which the rank is as minimal/maximal as possible

Find an $x$ in $\Bbb R$ for which rank of the matrix $$A=\begin{bmatrix}1 & 1&1&1 \\1 & -1&-1&1\\1 &-3 &-3 &x \end{bmatrix}$$ is as minimal/maximal as possible. I ...
1
vote
0answers
11 views

Derivation of the adjoint of a matrix

Let $V, W$ be vector spaces over any field $F$. A transformation $T:V \rightarrow W$, gives rise to the adjoint $V^* \leftarrow W^*:T^*$ of the dual spaces via: $$ T^*(f)(\cdot) = f\circ T(\cdot) $$ ...
0
votes
0answers
6 views

Clarifications regarding matrix transformations.

I have an equation which looks like this: Pos1 * L1 * X * L2 = Pos2 * R1 Where Pos1 and Pos2 are vectors. L1,X,L2 and R1 are matrices. I have to find the value for the matrix X. Please let me know ...
0
votes
0answers
8 views

Matrix problem in Mixed Regression

the background is $y= X \beta +e$ y=n*1 X=n*p $\beta=p*1$ e=n*1 take singular value decomposition of X $X=P \Delta Q$ $\beta=QKP'y$ K is a digaonal matrix and depending on its form can represent ...
0
votes
0answers
14 views

Binary Polynomial Factoring

I just need confirmation that I've done my math right. If $a(x) = x^4 + x^3 + x + 1$ and $b(x) = x^2 + x + 1$ are binary polynomials, find binary polynomials s(x) and r(x) such that $x^4 + x^3 + x + ...
1
vote
1answer
14 views
0
votes
0answers
4 views

Primitive elements of GF(8)

I'm trying to find the primitive elements of GF(8), the minimal polynomials of all elements of GF(8) and their roots, and calculate the powers of α^i for x^3 + x + 1. If i did my math correct, I ...
0
votes
0answers
11 views

What is the computational cost of reduced row echelon and finding the null space?

I'm taking computational linear algebra, and haven't been able to find too much information about the computational cost (in terms of m=rows and n=cols) of these two routines: Reduced Row Echelon ...
0
votes
2answers
30 views

Diagonalise without finding eigenvalues

I am asked to find the Jordan normal form (in this case, diagonalise) the $n\times n$ matrix $M$ defined: $$M_{ij}=1+\delta_{ij}\,x$$ I am then asked to deduce the minimal polynomial, eigenvalues and ...
0
votes
0answers
21 views

Linear algebra determinant-area relation question

I have an exercise where I am transforming a unit circle into an ellipse by some transformation $A$. Is it true that after the transformation the ellipse will have an area $\pi\cdot\mathrm{det}(A)$? ...
2
votes
3answers
18 views

Equation for a plane perpendicular to a line through two given points

The following type of question is quite popular with examiners at the institution where I study. Find an equation of the plane containing the point $(0, 1, 1)$ and perpendicular to the line passing ...
0
votes
0answers
12 views

Generator matrix of a Reed-Muller code [duplicate]

I need to find a generator matrix (2,4) of the Reed-Muller code (2,4), the dimension of R(2,4) and the minimum distance of R(2,4). I know that R(r,m) of order r, then length: n^m, dimension k = 1 + ...
1
vote
1answer
12 views

Complex matrix similar to a matrix with identical diagonal entries

Let $A$ be a complex matrix. Show that it is similar to a matrix with identical diagonal entries. I do have some sense, but could not prove it.
1
vote
2answers
23 views

Why inner product < , > on $C^n$ must satisfy the parallelogram law?

Why must norm induced by an inner product < , > on $C^n$ satisfy the parallelogram law? I know that there is a proof using $||v|| = \sqrt{(< v, v>)} $. But my concern is that why it still ...
1
vote
2answers
24 views

Positive definiteness of block matrices

I really appreciate if anyone can help me regarding my problem. I have a matrix in the format $M = \left[ {\begin{array}{*{20}{c}} {\delta I}&A\\ {{A^T}}&kA \end{array}} \right]$ where $A$ is ...
1
vote
0answers
21 views

Recurrence Derivative

I have a recurrence relation as follows $ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1(s)=sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2}(s)=\frac{s}{n+2}\{ ...
1
vote
1answer
21 views

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
0
votes
1answer
9 views

Properties of Hermitian and Positive Definite matrix

Let $A \in \mathbb{C}^{nxn}$ be Hermitian and positive definite. I have to show that $|a_{jk}|^2 < a_{jj}a_{kk}$ $max_{i,j=1,\dots,n}|a_{ij}| = a_{kk}$ for some $k$ with $1\leq k\leq n$ For ...
0
votes
0answers
34 views

Trace of a certain matrix

Let $A$ be a $227 \times 227$ matrix having distinct eigenvalues , with entries from $\mathbb Z_{227}$ , then what is the trace of $A$ ?
0
votes
2answers
41 views

Prove: If $T(u+kv) = T(u) + kT(v)$ then $T$ is linear

I was asked to prove this statement. Prove: If $T(u+kv) = T(u) + kT(v)$ then $T$ is linear It seems to me that for $k=1$ and $u=0$ the statement is proved. Is this correct? Many proofs use this ...
0
votes
2answers
19 views

Is there something called the Reduced Column echleon form?

I recently asked a question where I couldn't find the rank of a matrix. The question is : Problem on Finding the rank from a Matrix which has a variable At the time I believed in the answer, ...
0
votes
1answer
30 views

Why null space and column space?

I am not asking this question for WHAT is null space or WHAT is column space. I have finished learning about the definitions of these two concepts for a while. However, to install these concepts in my ...
0
votes
2answers
23 views

How to prove or understand this linear algebra assertion?

Given a matrix $B \in \mathbb{R}^{n \times k} $, and $B$ has rank $ k $. Therefore there exists a nonsingular matrix $A=( A_{1},A_{2}) \in \mathbb{R}^{n \times n} $ such that $$ AB= \left[ ...
2
votes
0answers
42 views

Shortest distance from a point to vertices of a cube

A $d$ dimension cube has vertices $P_1,...,P_{2^d}$, where the coordinates of each vertex are either $0$ or $1$. To find which vertex of $P_1,...P_{2^d}$ is closest to a given point $P=(p_1, ...
2
votes
1answer
18 views

How to find $α^2(β^4 +γ^4 +δ^4)+β^2(γ^4 +δ^4 +α^4)+γ^2(δ^4 +α^4 +β^4)+δ^2(α^4 +β^4 +γ^4)$

How to do the part (iv) . Please help. Here are my answers to the first parts: (i) α a root of given equation $\implies \alpha^4-5 \alpha^2 + 2 \alpha -1 = 0$ $\implies \alpha^{n+4} - 5 ...
3
votes
0answers
30 views

Isometry problem [on hold]

An isometry $M : R^n → R^n$ is a map that preserves distance, i.e. $||M(v) − M(w)|| = ||v − w||$ for all $v, w ∈ R^n$ . • Let M be an isometry with $M(0) = 0$. Let $e_i$ be the $i$th standard unit ...
1
vote
1answer
22 views

Any invertible linear map is homotopic to a composition of reflections

I am trying to solve a problem in Hatcher. I reduced my problem to showing that if $f:\mathbb{R}^n\to\mathbb{R}^n$ is an invertible linear map, then $f$ is homotopic to a composition of reflections, ...
0
votes
0answers
31 views

Proving $A_{n}$ is not invertible for n>2 when the entries are sequential integers

Let $A_{n}$ be the nxn matrix whose entries are the integers 1, 2, 3,..., n-1, n, written in order from left to right, top to bottom. For example, $$A_{5}=\begin{bmatrix} ...
1
vote
1answer
28 views

$\operatorname{tr}(AB) = 0$ for (skew-)symmetric matricies

I know if A is symmetric and B is skew-symmetric then $\operatorname{tr}(AB) = 0$. (This follows because $\operatorname{tr}(AB) = -\operatorname{tr}(AB) $) Is the converse of that true? In other ...
0
votes
1answer
33 views

If A^2 =0 then possible rank of A

Let, A be a non zero matrix of order 8 with A^2 =0. Then one of the possible value for rank of A is (a) 5 (b) 4 (c) 6 (d) 8
0
votes
0answers
15 views

Trying to show that conic section $ax^2 + 2bxy + cy^2 = d$ takes on different shapes.

Note: This is a homework problem. I'm trying to show that conic section $ax^2 + 2bxy + cy^2 = d$ is an ellipse or the empty set if $ac-b^2\gt 0$. There are others to show but if I can understand this ...
1
vote
0answers
30 views

Partial Sum to be invertible [on hold]

Let $A_1,\cdots,A_m$ be $n\times n$ matrices, satisfying $$m>n, A_1+\cdots+A_m=E_n,$$ where $E_n$ is the $n\times n$ identity matrix. Show that there exists a subset $P\subset \{1,\cdots,m\}$ ...
3
votes
2answers
41 views

Why is $\det(A-\lambda I)=(\lambda-c)^n$ when $(A-cI)^n=0$?

Let $A$ be a $n\times n$ matrix and suppose that $(A-cI)^n=0$ for some scalar $c$. Then why the characteristic polynomial of $A$ is $(x-c)^n$?
1
vote
3answers
27 views

Compute $B=QAQ^{-1}$

$A,B$ are $n\times n$ matrices, $B=QAQ^{-1}$, and I know $A$ and $B$, how to compute $Q$? I know if $T$ a linear transformation, and with different basis we get $A$ and $B$, and we could use these ...
0
votes
0answers
7 views

Number of pivot columns in a 4x6 matrix for spanning set to occur

How many pivot columns must a 4x6 matrix have if its columns span $\mathbf{R}^4$? Explain. So, in my head, this is pretty clear: You need four dimensions => So you need a minimum of four vectors that ...
1
vote
2answers
53 views

Is the Square Root of an Inverse Matrix Equal to the Inverse of the Square Root Matrix?

I know in general that if a matrix $A$ is positive definite, then there exists a (unique?) square root matrix $B$, which is also positive definite, such that $BB=A$. Therefore, suppose $A$ is ...
0
votes
1answer
22 views

Linear Algebra Analytical Exercise

This one has me stumped... $$H=C(sI-A)^{-1}B$$ and $$H_{CL} = C(sI-A+BK)^{-1}BG$$ Show that $$H_{CL} = H[I+K(sI-A))^{-1}B]^{-1}G$$ Any hints would be greatly appreciated!
-4
votes
0answers
35 views

Find a basis for symmetric $2 \times 2$ matrices [on hold]

Find a basis for the space of all $2 \times 2$ symmetric matrices. I do not even know how to start. please explain it to me step by step
-1
votes
1answer
19 views

Solving symbolic linear equations with maple

How can I solve linear equations of the following type in Maple? $$\begin{pmatrix} 1 & 1 & 1 & 1\\ b-c & c-b & a-b &0 \\ b-d & d-a & 0 &a-b \end{pmatrix} ...
0
votes
0answers
9 views

Eigenvalues and Positive-Definiteness of the Hessian Matrix

Suppose we have a function $f \in C^{2}$ and the Hessian defined as follows: $Hf(x,y)(h) = \displaystyle\frac{1}{2} \begin{pmatrix} h_{1} & h_{2} \\ \end{pmatrix} \begin{pmatrix} f_{xx} ...
1
vote
1answer
22 views

Simple Eigenvalue finding question (by gauss elimination)

I saw a method for finding eigenvalues by using Gauss elimination to find an upper triangular matrix, then just taking the diagonal elements as the eigenvalues. It seems to work except for this case: ...
-2
votes
0answers
13 views

Subspace vector proofs problem [on hold]

I'm having trouble understanding/solving this proof. QUESTION: Prove the set P_3 is a subspace of P_4 with standard operations, where P_n is a vector space of all polynomial functions with degree n ...
1
vote
1answer
23 views

Basis for vector space $\mathbb{R}^{m\times n}$

My question is whether my solution to the following problem is valid. The problem is from Artin's Algebra, chapter 3: Let $(X_1,\cdots,X_m)$ and $(Y_1,\cdots,Y_n)$ be bases for $\mathbb{R}^m$ and ...
0
votes
1answer
24 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
2
votes
2answers
32 views

What does determinant of linear operator mean?

I am solving problem (Linear Algebra by Hoffman, Excercise 5.4.8) : Let $V$ be the vector space of $n\times n$ matrices over the field $F$. Let $B$ be a fixed element of $V$ and let $T_B$ be the ...
1
vote
2answers
16 views

Representation of Matrix with Rank 1

Prove that every $m \times n$ matrix of rank $1$ has the form $A=XY^t$, where $X,Y$ are $m$- and $n$-dimensional column vectors. How uniquely determined are these vectors$?$ My attempt: I thought ...
1
vote
0answers
47 views

Is it true that $V$ and $V^*$ are naturally isomorphic as finite vector spaces if $V$ is equipped with an inner product?

This is a homework question from my differentiable manifolds class: In general we know that if $\dim V<\infty$ then $V$ and $V^*$ are isomorphic because any two vector spaces with the same ...