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

1
vote
1answer
38 views

Why should there be a 7-dimensional cross product in the context of exterior algebra?

The three-dimensional cross product can be viewed as the wedge product corresponding to the exterior power $\Lambda^2(\mathbb R^3)$. An explanation that I have come up with for the scarcity of cross ...
0
votes
0answers
23 views

Fast method for getting solution for underdetermined equation system

What is a fast and stable method for getting a solution for an underdetermined equation system which could be applied by a computer?
0
votes
0answers
32 views

algorithm for generating a random non-degenerate matrix over $[0,1 ]$?

I want to generate a random matrix $V\in [0,1]^{(n+1)\times n}$(not necessarily being binary ), such that for each row of $V$, there is at least one component is $1,$ and at least one component is ...
2
votes
1answer
44 views

range of $m$ such that the equation $|x^2-3x+2|=mx$ has 4 real answers.

Find range of $m$ such that the equation $|x^2-3x+2|=mx$ has 4 distinct real solutions $\alpha,\beta,\gamma,\delta$ To show how I got the wrong answers. From $|x^2-3x+2|=mx$ I got the two case ...
0
votes
1answer
11 views

linear transformation sequences

I'm working on this exercize about linear transformation: Let $E=\mathbb{R^N}$, $T:E \rightarrow E: \ (u_n)_{n\geq 0} \rightarrow (u_{n+1})_{n\geq 0}$ $S:E \rightarrow E: \ (u_n)_{n\geq 0} ...
5
votes
1answer
51 views

Is there a problem in assuming that a point is the same thing of a vector?

I've read Apostol's Calculus, in the section on analytic geometry. He says that he's going to use 'vector' and 'point' interchangeably. But in Beardon's Algebra and Geometry, he argues that there is ...
1
vote
2answers
34 views

Prove $rank(BA)=rank(A)=rank(AC).$ where $A\in F^{mXn}$ and $B\in F^{mXm}$ and $C\in F^{nXn}$ be invertible matrices

This is my endsem question and i am stuck at it.Question is like this. Let $A\in F^{mXn}$,Let $B\in F^{mXm}$ and $C\in F^{nXn}$ be invertible matrices.Prove that $rank(BA)=rank(A)=rank(AC).$ How to ...
1
vote
0answers
12 views

Understanding the Frobenius Norm for Sparse Coding

I have a question regarding sparse coding, Non-negative sparse coding. Iterate until convergence: $ \mathbf{A_i} \leftarrow \arg \! \min_{A \geq 0} || \mathbf{X}_i - \mathbf{B}_i\mathbf{A}||_F^2 + ...
-3
votes
0answers
55 views

How Many Days Until 100 Screens [on hold]

I earn 2 dollars each day from a screen costing 200 dollars.In how many days can I buy 100 screens if invest back all my earnings to buy new screens? To start with I have $200 in hand and no ...
2
votes
3answers
506 views

Does matrix addition give you a matrix or a number?

I am very confused by something our lecturer said today: We were given two matrices: $B=\begin{pmatrix}2 & 3\\ 2 &0 \\ 0&3\end{pmatrix}$ C=$\begin{pmatrix}6 ...
0
votes
1answer
18 views

Length of a projected line

If a line is of true length x and is inclined in angles a,b,c with respect to the xy,yz,zx planes respectively , then how can i find the length of the projected line in the xy , yz and zx planes ...
0
votes
1answer
21 views

identity operator, direct sums, and projections

Let W be finite-dimensional vector space. Let $P: W\to W$ be a projection. Let U = Range(P) and V=Ker(P) (a) show that P is the identity operator on U. I dont understand the problem ...
0
votes
2answers
16 views

Invariant subspaces of the identity map

How many invariant subspaces does the identity map on $R^2$ have? My attempt: {0}, which coincides with the kernel. $R^2$, which coincides with the image and eigenspace. Is that it?
1
vote
1answer
24 views

Product of orthogonal projections

I need an example of two orthogonal projections such that their product is not a projection. I'm aware of this: Product of orthogonal projections need not be a projection Unfortunately, I've no idea ...
1
vote
1answer
25 views

$ det(A).\lim_{t \to 0} \frac{det(Id+tA^{-1}X)- det(Id)}{t} =tr (A^{-1}X)$

In some note it is written that $$ det(A).\lim_{t \to 0} \frac{det(Id+tA^{-1}X)- det(Id)}{t} =tr (A^{-1}X)$$ I could not understand how this is happen. Can someone explain it in detail please.
0
votes
0answers
8 views

Finding the decomposition of a function on a cubic spline basis of functions

In a computational project, I need to solve a partial differential equation. Standard procedure is to consider the weak formulation of the problem which maps it onto an algebraic problem. With cubic ...
0
votes
1answer
25 views

Is there a difference between linear map and linear transform?

Wikipedia page says there is no difference but when I see reference to a map it's domain and range are specified but in a transform it is not the case. Any suggestion how to view the both?
0
votes
0answers
14 views

runninig pseudoinverse of a wide possibly rank-deficient matrix

There is a wide matrix $A=[a_1|a_2|..a_n]$ with m rows and n columns where $m<n$ where $a_i$ is the $i_{th}$ column vector of $A$. Let $A'$ be $A=[a_2|..a_n|a_{n+1}]$. Is there an efficient way of ...
0
votes
1answer
11 views

Finding individual vectors given a dot product or vector length

$$u \cdot v = 2\\ v \cdot w = -6\\ u \cdot w = -3\\ ||u|| = 1\\ ||v|| = 2\\ ||w|| = 7\\$$ (a) $<2v - w, 3u + 2w>$ If I'm given a vector length or dot product like the above how can I find the ...
3
votes
1answer
27 views

What's the best way to think about the covariance matrix?

Let $X$ be a random vector with covariance matrix $\Sigma$. People often describe $\Sigma$ in terms of its components: $\Sigma_{ij}$ is the covariance of the $i$th and $j$th components of $X$. But ...
1
vote
1answer
22 views

Can we treat Covariance matrix as linear transformation.

This question is related to the link: What is the difference between matrix theory and linear algebra? My question is: when we see a matrix in any equation how to determine if that matrix is a ...
1
vote
0answers
30 views

Permutation as a product of generators of the permutation group

Let $G$ be a permutation group, generated by $g_1,\ldots,g_n$. And let $h$ be in $G$. Example: $G=\langle (12)(34),(123)\rangle$ and $h=(12)(34)(123)=(243)$ (reading the cycles from right to left, ...
0
votes
1answer
39 views

Finding a pair of Orthogonal Vectors

Want: Pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3) My attempt at a solution: I got stuck...
0
votes
1answer
21 views

Let W be the collection of all 2 by 2 symmetric matrices. Describe the orthogonal complement of W. (please)

A matrix is symmetric if $A^T$=A And the standard basis for symmetric matrices is [a,b], [b c] written as rows of a 2x2 matrix (sorry don't know how to make a matrix on this site). My question: How ...
0
votes
1answer
22 views

prove that a system $AX=Y$ has soloutions iff the row rank of A is same as that of the augmented matrix?

Let A be a m by n matrix over the field F , I want to prove that a system $AX=Y$ has solutions iff the row rank of A is same as that of the augmented matrix of the system My try: I was thinking that ...
0
votes
1answer
40 views

Linear Algebra by Friedberg Chapter 1.3 Problem 28

I know how to show that $W_1$ is a subspace but I don't know what it wants me to think by saying "assume $F$ is not of characteristic $2$". I know that when $F$ is not of characteristic $2$, it ...
3
votes
0answers
29 views

Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
1
vote
0answers
24 views

Eigenvalues and eigenvectors for earthquake modeling

My instructor explicitly stated that, because we are asked to find eigenvalues and eigenvectors of a $7\times 7$ matrix, MATLAB would be easiest to use. The equation $(1)$ is intended to resemble ...
0
votes
1answer
17 views

Effects of Isomorphic Transformations on Vector Spaces.

Let $V$, $W$ be finite-dimensional vector spaces and let $T: V\rightarrow W$ be an isomorphism. Let $X$ be a subspace of $V$. Show that $T(X)$ is a subspace of $V$. My attempt: I know two vector ...
3
votes
3answers
44 views

Rotate an area around a diagonal line.

I know how to find the volume of the figure formed when you rotate a $2$-dimensional area around a horizontal or vertical line, but what if it were a diagonal line instead? For example: Rotate the ...
2
votes
0answers
26 views

What is the volume inside $S$, which is the surface given by the level set $\{ (x,y,z): x^2 + xy + y^2 + z^2 =1 \}$?

The solution given uses a linear algebraic argument that doesn't seem very instructive -- and may not even be correct, I think. We notice from the equation, that the surface is a quadratic form, ...
1
vote
3answers
28 views

Volume of a parallelepiped when not given values for three vectors

There is a parallelepiped determined by three dimensional vectors x, y, and z. The volume of this parallelepiped is $11$. What is the volume of the parallelepiped determined by the three dimension ...
2
votes
1answer
17 views

About the Affine hull and Span.

I'm learning Linear Algebra and Convex Optimization simultaneously, I notice that the affine hull is, to some extent, analogous to the span, but when I read the lines "We define the affine dimension ...
1
vote
1answer
51 views

What kind of vector spaces have exactly one basis?

It's an exercise in chapter 2 of the book Linear Algebra Done Right.
1
vote
2answers
28 views

Find a basis for the row space and a basis for the column space

By inspection, find a basis for the row space and a basis for the column space for the following matrix: $$ \begin{bmatrix} 1 &2& -2& 7 \\ 0 &1& 3 &5 \\ 0& 0& 1 ...
2
votes
3answers
95 views

Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A = \sum_{i=1}^n f_0(x_1, \ldots, Ax_i,\ldots, x_m)$

Question: Consider $f: (-\epsilon, \epsilon) \to \mathbb R^{m^2}$ a differentiable path of matrices $m \times m$ such that $f(0) = I_m$ and the function $g: I \to \mathbb R$ is defined by $$g(t) = ...
2
votes
1answer
46 views

Degree of minimum polynomial at most n without Cayley-Hamilton?

Let $T$ be a linear transformation of an $n$-dimensional vector space $V$ over a field $k$. It's pretty easy to define the minimum polynomial of $T$ and make sure its degree is between $1$ and $n^2$, ...
1
vote
1answer
31 views

dim of quotient spaces

Consider $V=Mat_n(K), n \ge 1$ and 3 subspaces $U_1,U_2,U_3 \subset V$ respectively the scalar, diagonal and upper triangular matrices. Calculate dimension of $U_3/U_1 \ \ U_3/U_2 \ \ V/U_1 \ \ ...
1
vote
1answer
63 views

Is this special matrix invertible?

The symmetric, tridiagonal $n-$by$-n$ matrix with the elements $a_{ii+1} = a_{i+1i} $ and off-diagonals' absolute values equal to the diagonal (except for row 1 and row n) is invertible. The elements ...
0
votes
1answer
27 views

Show that the Gauss ring is Euclidean ring

Show that the Gauss ring $ (\mathbb{Z}[i]=\{a+bi \mid a,b \in \mathbb{Z}\},+,\cdot ) $ is Euclidean ring with a norm $d(a+bi)=a^{2}+b^{2}$ How to prove that theorem? I started checking the first ...
0
votes
1answer
29 views

Coefficients of a linear combination

Suppose I have a vector space $V$, with $\{\vec e_1, \ldots, \vec e_n\}$ forming a basis of $V$. Also, suppose that I have a vector $\vec x \in V$, which I want to express as the unique linear ...
1
vote
2answers
25 views

What is the relationship between dimension of eigen space and multiplity of eigen value?

Is there a relationship between dimension of eigen space with respect to an eigen value $\lambda_i$ and multiplicity of eigen value $\lambda_i$ ( by multiplicity I mean if $(\lambda-2 )^3(\lambda ...
2
votes
2answers
23 views

how to combine angle rotations along different axes into one rotation along a single vector [duplicate]

So, lets say I have some rotation a about the x-axis(vector:$(1, 0 ,0)$) and some other rotation about y-axis(vector $(0, 1, 0)$) and a rotation about the z-axis(vector: $(0,0,1)$). How would I ...
2
votes
1answer
60 views
+200

Why a form is positive only if its matrix in some ordered basis is a positive matrix?

I'm reading Hoffman's "Linear Algebra" Chapter 9 "Operators on Inner Product Spaces" and got lost at the positive property on (sesqui-linear) forms, operators and matrices. The confusing comes from ...
1
vote
0answers
30 views

Non-unique factorization in $\mathbb{Z}[\sqrt{-5}]$

I want to show that the decomposition into irreducible factors in the ring $$\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5}|\space a, b \in \mathbb{Z}\}$$ is not unique, except for the order of factors ...
1
vote
1answer
15 views

Proof if $A$ is normal then it is nondefective

What is the proof that if $A$ ($m\times m$ Matrix) is normal i.e $(AA^{\ast} = A^{\ast}A)$ then $A$ is non defective i.e (for each eigenvalue of $A$, its algebraic multiplicity is equal to the ...
2
votes
4answers
41 views

If a real linear operator $T$ satisfies $T^{t}T = TT^{t}$, is it necessarily true that $T = T^{t}$?

If $T^{t}T = TT^{t}$, does it imply that $T = T^{t}$? Here, $T$ is a linear operator on a real vector space.
1
vote
1answer
27 views

Showing $T:K^n \to K^{n-1}$ is surjective

Hi everyone, I'm a bit stuck on this question. Could anyone share some ideas? Note: $K$ is the field I believe from the definition of the $ker(T)$ we can tell $n = 3$, but I am unsure as to how ...
0
votes
0answers
13 views

How to visualise the step where $T(\mathbf{e_j})=\sum_i^n{a_{ij}\mathbf{f_i}}$?

Consider a linear map $T:\mathbb{F}^n \rightarrow \mathbb{F}^m$ and let $\{\mathbf{e_j}:j\in[1,n]\}$ be the standard basis for $\mathbb{F}^n$,$\{\mathbf{f_i}:i\in[1,m]\}$ be the standard basis for ...
1
vote
1answer
16 views

Polynomial Factorisation - Linear Algebra

Im attempting a linear algebra question in which I have been given the following quadratic form $q(x,y,z) = x^2+25y^2+10xy+2yz$. I have to find a basis $B$ such that $[f]_B$ has the real canonical ...