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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2
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2answers
24 views

Definition of an image of a linear transformation

I have the following definition of an image of a linear transformation, but I think that there's a mistake in the size of the field (confusion between m and ...
0
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0answers
20 views

Find the image of a vector

I have an endomorphism and I have to study the image of a vector $v$. How can I do this? Can you please give an example? I know how to calculate the inverse image but I have many doubts on the ...
0
votes
0answers
22 views

Is there a relation to determine condition (positive or negative definite) of C, if C = A+B and A, B are positive and negative definite?

I have a question: Matrix A and B are positive and negative definite, respectively. Is there a relation to determine whether C is positive or negative definite, if C = A+B?
5
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3answers
56 views

If $X + X^T$ is positive definite, is $X^{-1} + X^{-T}$ also positive definite?

Is it true or is there a counterexample?
0
votes
0answers
9 views

$\frac{4}{3}(1,-1,0)^T+\textrm{vect} \left((-1,1,-1)^T\right)$

Here I have a passage on the conclusion, but I do not understand it. the conclusion say : $f$ is the oriented axis of screw.. how I can from $\frac{4}{3} \begin{pmatrix}1\\1\\0 ...
0
votes
1answer
33 views

How to bound the biggest eigenvalue of $\sum_{i=1}^{n}x_ix_i^T$?

My question is to bound the biggest eigenvalue of $A=\sum_{i=1}^{n}x_ix_i^T$, where $x_i\in\mathbb{R}^d$ is a column vector. My idea is, to bound the biggest eigenvalue of $A$, i.e. $\|A\|_2$. I can ...
0
votes
0answers
21 views

What is a “supplementary subspace”?

Let $Q$ be a quadratic form of vector space $V$ over a field $k$ with characteristic $\neq 2 $, $V^{0}$ be its orthogonal complement. If $U$ is a supplementary subspace of $V^0$ in $V$, then $V = ...
0
votes
0answers
22 views

Reference for the proof of interlacing of eigenvalues of submatrices

If one has a $n \times n$ Hermitian matrix $A$ and one removes $k$ of the rows and their corresponding columns then the eigenvalues of the remnant interlace the eigenvalues of the full matrix. Can ...
0
votes
3answers
46 views

Show that $\operatorname{span}(\operatorname{span}\{\vec{x},\vec{y}\}) = \operatorname{span}\{\vec{x},\vec{y}\}$

If I am given fixed vectors $\vec{x},\vec{y}\in \Bbb R^n$, how can I show that $\operatorname{span}(\operatorname{span}\{\vec{x},\vec{y}\}) = \operatorname{span}\{\vec{x},\vec{y}\}$? I am a little ...
0
votes
0answers
13 views

extract the vector X in an equation

For the following question I need to extract the X from the equation $A = \left[ {B_{0,0}^TX, \ldots ,B_{0,M - 1}^TX, \ldots ,B_{N - 1,0}^TX, \ldots ,B_{N - 1,M - 1}^TX} \right]$ where $X$ and ...
1
vote
2answers
30 views

Algebra QF $f(x)=x^2-18x-4$

I have the problem to find the zero of the function: $$f(x)=x^2-18x-4$$ I have it mostly worked out as $a=1, b= -18, c= -4$ worked out I have: $$\frac{18\pm \sqrt{340}}{2}$$ I know the answers are ...
0
votes
1answer
13 views

Rank of matrix in relation to number of rows and columns

From my linear algebra text: “Suppose that rank $A = r$, where $A$ is a matrix with $m$ rows and $n$ columns. Then $r < m$ because the leading 1s lie in different rows, and $r < n$ ...
0
votes
1answer
24 views

Is every finite dimensional linear space a banach space [on hold]

Is every finite dimensional linear space a Banach Space? Is every finite dimensional linear space a Hilbert Space?
3
votes
2answers
48 views

The purpose of LU Decomposition

I was curious if anyone could help me understand why an LU decomposition is useful from a theoretical or computational standpoint. It seems to me that it is just a way to teach students the basics of ...
0
votes
1answer
27 views

Graphing linear, affine, and convex combinations

For the vectors (2, 1) and (1, 3), how would I graph each of the three combinations? Here are my thoughts (sorry might be totally wrong): linear - plane connecting the two points affine - infinite ...
0
votes
1answer
22 views

Given a square matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues?

Given a matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues? If instead the matrix had its nonzero entry component at ...
1
vote
0answers
23 views

How to determine if a set represents a line, plane or hyperplane?

How do you approach a question that gives you a set and asked to determine if it represents a line, plane or hyperplane? The Question: https://www.dropbox.com/s/0gscqur18kqg3ma/SpanningQuestion.PNG ...
2
votes
0answers
24 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
1
vote
1answer
27 views

Using absolute value to solve an equation.

I am required to have two answers to this problem: A 'Larger Number' and a 'Smaller Number'. I know that I have half of the question correct (The 'Larger Number' answer). But I do not know how to ...
0
votes
2answers
57 views

Matrix invertible iff det(matrix)$\neq 0$?

When we want to find the inverse of the matrix $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ we're searching for a matrix $$\begin{bmatrix}x & y \\ z & w\end{bmatrix}$$ such ...
1
vote
0answers
38 views

prove the subspace is equal to L1∩L2

Let $L1,L2$ be subspaces of $V$ and $dim(L1+L2)=1+L1 \cap L2$ Show that $L1 \subseteq L2$ or $L2 \subseteq L1$ and $L1+L2= L1$ or $L2$ it's has something to do with if $L1,L2 \ne L1 \cap L2$ ...
-5
votes
0answers
29 views

Find value of x from the span of vectors. [on hold]

Find 'x' such that the vector is the span of set of vectors:
0
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1answer
32 views

Relationship of basis vectors of the complex plane

I am working on learning more about the connection of complex numbers and rotations in the context of rational geometry. Thanks ahead of time for any corrections on my best assertions. Let $B$ ...
1
vote
1answer
22 views

Triangularization of a matrix.

so I need to find an invertible matrix $P$ such that $P^{-1}AP$ is upper-triangular, where $$A = \begin{bmatrix} 4 & -1 \\ 9 & -2 \end{bmatrix}$$ So I found that the eigenvalue is $1$ which ...
-4
votes
0answers
22 views

By using the properties of determinants prove that [on hold]

By using the properties of determinants, prove that:
0
votes
1answer
30 views

About Lines and Planes in Linear Algebra

The set of linear combinations $cX$ is said to span a line. How can I see that there's correspondence between $cX$ and the equation of a line $ax + by = 0$? Besides, the aforementioned equation looks ...
0
votes
0answers
33 views

Gradient function

Let A (red) and B (green) 2 distinct points anywhere in a 3D space. I am looking for a function which take a point P, and returns the value in blue in the picture. Each blue number in the picture ...
2
votes
4answers
60 views

Linear dependence of $\left\{x^{n}\,\colon\, n\in\mathbb{N}\right\}$

Consider the set $S=\left\{x^{n}\,\colon\, n\in\mathbb{N}\right\}$. (Note that $x\in\mathbb{R}$) Is this set linearly dependent? Well thinking about it we want to find some non-trivial values ...
1
vote
2answers
55 views

Can a matrix have eigenvalue with infinite multiplicity?

Suppose we have matrix of the form $$ A= \begin{bmatrix} a & -1 \\ 0 & a \\ \end{bmatrix} $$ and we would like to analyze its diagonalizability. By taking the ...
0
votes
1answer
20 views

Subspaces of a functional space

Suppose $(V,\,\oplus,\,\odot)$ is a vector space where $V$ is the set of all functions $f\colon\mathbb{R}\to\mathbb{R}$ and the operations $\oplus$ and $\odot$ are defined by $$f\oplus g = x\mapsto ...
0
votes
1answer
23 views

AB = Identity matrix; matrices; determinants; proof

Let $M(n\times n, \mathbb Z)$ be the set of all $n\times n$- matrices with integer coefficients, and a matrix $A \in M$. Proof, that: There is exactly one matrix $B \in M(n\times n, \mathbb Z)$ with ...
0
votes
0answers
12 views

Inequality involving expectations of vector/matrix norms

I'm reading a paper and trying to understand the proof of a lemma regarding expectations of norms of random vectors. The author's notation does not distinguish between vector and matrix norms, nor ...
0
votes
3answers
36 views

Why has the space $\{X\in M(3,\mathbb{R}) : X+X^T=0\}$ dimension $3$ over $\mathbb{R}$

How can we determine this space $\{X\in M(3,\mathbb{R}) : X+X^T=0\}$ is $3$ dimensional over $\mathbb{R}$. Here I can find a linearly independent set which has $3$ elements. So I know the dimension is ...
0
votes
0answers
28 views

An integral with respect to the Haar measure over the unitary group

I am trying to find the answer of this integral: $$E:=\int dU \ (U^2 \otimes I) M (U^{ \dagger 2}\otimes I) $$ That is an integration with respect to the Haar measure and $U$'s are $d\times d$ ...
0
votes
1answer
12 views

dimension of an intersection of subspaces

Let $V$ be the vector space of all polynomials in one variable with real coefficients having degree at most 20. Define the subspaces \begin{align*} W_1 &=\{p \in V; p(1)=0,p(1/2)=0, ...
1
vote
1answer
26 views

how to find the solution of this cost function?

I have the following cost function. $J = \sum_{i=1}^N a\, Trace(W^TX_iW) - b\, Trace(W^TY_iW)$ Where $X_i$ and $Y_i$ are symmetric matrices, $a$ and $b$ are scalars. How can I find W?
0
votes
1answer
36 views

$\operatorname{rank}(A) = $max number of rows of submatrix $B$; Proof

I don't understand how to proof the following: The rank of a matrix $A \in M$ ($m \times n$, Field) equals the maximum number of rows of a square submatrix $B$ of $A$ with $\det (B) \neq 0$. The ...
1
vote
2answers
22 views

Different Representation of Linear System

Currently working on my Linear Algebra homework and I have come across several problems of the same form that I haven't the slightest clue how to approach. Keep in mind I haven't taken math classes ...
-1
votes
0answers
21 views

How can I prove that the set of $A$-invariant subspaces forms a lattice?

How can I prove the following proposition: the set of $A$-invariant subspaces forms a lattice.
0
votes
1answer
36 views

Trying to define a simple “warp” function

I'm trying to define a 2D "warp" function y=f(x,w). A picture is worth a thousand bytes: I am looking for a simple function f(x, w) that satisfies the ...
1
vote
3answers
176 views

To show two matrices are conjugate to each other

Given two matrices A and B $$ A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 0 & 4 \\ 0 & 1 & 0 \\ 0 ...
1
vote
1answer
21 views

The inverse of the sum of two matrices in *Applied statistical decision theory *.

I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here. A theorem at the page linked states that if two matrices $A,B$ are non-singular and of ...
0
votes
1answer
46 views

To find a linear transformation such that $T^{2}( v) = -v $

Question is to find a linear transformation from $R^{2}$ to $R^{2}$ such that $T^{2}( v) = -v $ for all v .I used hit and trial method to do this but its has been a long time i am stuck at it .Is ...
0
votes
0answers
39 views

Combinatorial approach to calculate determinant

Suppose you have set of $n*n$ matrices with entries from the set $\{1,-1\}$. Then what can be the maximum determinant which you can obtain from such type of matrices.
4
votes
3answers
71 views

Determinant of the inverse matrix [duplicate]

I'm seeking for a proof of the following: Let $A$ be an invertible matrix. Then the determinant of $A^{-1}$ equals: $$\left|A^{-1}\right|=|A|^{-1} $$ I don't know where to begin the proof. Any ...
0
votes
1answer
29 views

Eigen value system? solution

I have the following system. $AW = \lambda B W$ Where $A,B,W$ are matrices and $\lambda$ is a scalar. The values of $A,B$ and $\lambda$ are known. $B$ is invertible. This is a solution to an ...
1
vote
1answer
28 views

linear functionals linearly independent

Let $V$ be a vector space with $\dim V=n$. Let $\varphi_1,...,\varphi_n $ be linear functionals that are not $0$. Prove that $\varphi_1,...,\varphi_n $ are linearly independent if and only if ...
11
votes
5answers
561 views

Studying math all day and really young [on hold]

I am very young and want to learn algebra and calculus for fun. What should I keep in mind when I start learning? I am going to try the textbooks I have borrowed out: Dummit and Foote and Spivak's ...
-1
votes
1answer
25 views

Set of linear transformation with T (1,0,1)=(1,2,3), T (1,2,3)=(1,0,1) [on hold]

Let $$S=\{T:{\mathbb R}^3\to {\mathbb R}^3\}|\quad T \quad \text{is linear transformation with}$$ $$\quad T (1,0,1)=(1,2,3),\quad T (1,2,3)=(1,0,1)\}. $$ Then $S$ is Singleton set Finite set having ...
2
votes
1answer
29 views

How to prove this result using Permutations? [on hold]

Let A be the set of all $3*3$ skew symmetric matrices whose entries are either -1, 0 or 1. If there are exactly 3 zeroes, three 1's and three (-1)'s, then prove that only 8 such matrices can exist.