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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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 ...
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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 ...
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1answer
24 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 ...
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2answers
45 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 ...
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27 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 $L1,L2 \ne L1 \cap L2$ ...
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1answer
12 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:
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17 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 ...
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1answer
15 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 ...
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20 views

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

By using the properties of determinants, prove that:
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1answer
27 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 ...
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26 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 ...
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4answers
43 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 ...
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2answers
52 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 ...
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1answer
18 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 ...
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1answer
21 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 ...
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0answers
7 views

Inequality involving expectations of vector/matrix norms

I'm reading a paper and trying to understand the proof of a simple lemma regarding expectations of norms of random vectors. The author's notation does not distinguish between vector and matrix norms, ...
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3answers
32 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 ...
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15 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 $d\times d$ ...
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1answer
9 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, ...
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1answer
14 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?
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1answer
34 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 ...
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2answers
20 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 ...
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12 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.
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1answer
25 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 ...
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3answers
166 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 ...
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does a closed form solution exist for this equation?

I have a cost function $J$, which depends on a projection matrix $W$, which is unknown. When I get the partial derivative $\frac{\partial J}{\partial W}$ the equation is: $\frac{\partial J}{\partial ...
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1answer
16 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 ...
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1answer
39 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 ...
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0answers
33 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.
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3answers
66 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 ...
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1answer
26 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 ...
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1answer
24 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 ...
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5answers
500 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 ...
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1answer
22 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 ...
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1answer
28 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.
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61 views

Where can I find linear algebra described in a pointfree manner?

Clearly, some of linear algebra can be described in a pointfree fashion. For example, if $X$ is an $R$-module and $A : X \leftarrow X$ is an endomorphism of $X$, then we can define that the ...
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2answers
35 views

Change of Basis for $2\times2$ matrix

Suppose I have the matrix basis $\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&1\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&0\\1&0\\\end{bmatrix}$, ...
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1answer
23 views

Prove the following vectors are linearly independent

So I have these three vectors: [i, 2+i, 3]; [2, -i, 4-i}; [3, -1, 2] and I need to show they are linearly independent. This means that given scalars $x_1, x_2, x_3$ their scalar sum should equal 0. ...
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2answers
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how many jelly beans did each girl have at first?

Martha and Mary had $375$ jelly beans in all. After Mary ate $24$ jelly beans and Martha ate $\frac 17$ of her jelly beans, they each had the same number of jelly beans left. How many jelly beans did ...
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28 views

Quention about the historical definition of determinant

$$ax+by = k_1\\cx + dy = k_2$$ If I want to solve for $y$ in the first equation: $$by = k_1 - ax\implies y = \frac{k_1-ax}{b}$$ Then substitute $y$ in the second equation: $$cx + d\frac{k_1-ax}{b} ...
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1answer
27 views

Trouble showing spans of two bases are equivalent

I was given the following problem: Let V be a vector space over field F. Show that x,y $\in$ V form a basis iff x+y, x-y form a basis. But I seem to be stuck when showing the span of one basis ...
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1answer
33 views

Finding a linear transformation such that $T^{3} = T $

I have to show that there exists a linear transformation such that $T^{3} = T $ i can see that from here that T has eigen values $0.1.-1$ .But how do i find linear transformation .Also for v and q ...
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2answers
45 views

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, Prove that $|(A+A^T)(B+B^T)|=0$

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, prove that $\det[(A+A^T)(B+B^T)]=0\ \ $ with $ \ k\in \mathbb{N}$ I don't have ideas for this problem. Thanks !
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Determinant over $\mathbb{C}$ of an $\mathbb{H}$-linear mapping.

Let $V = \mathbb{C}^n$ and let let $u$ be a $\mathbb{C}$-linear endomorphism of $V$. Then $u$ can also be considered as an $\mathbb{R}$-linear mapping $u_{\mathbb{R}}$. It is well known that $$\det ...
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21 views

Representing commuting operators as functions of a third operator.

Let A and B be commuting operators, then there is a maximal operator C and there are functions f, g such that A = f(C) and B = g(C). I'm looking for a proof of this theorem. I don't fully understand ...
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1answer
24 views

Prove or disprove isomorphism problem

P is a 2*4 matrix, which has rank (P) = 2, L: M 4*4 -> M 2*2 is a linear mapping, defined by L(A) = P A P^T, ---(PAP transpose). I can see that L is not one-to-one, as A must be in the null-space of ...
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2answers
24 views

Finding a Basis for polynomial subspace

This is problem 14 in Herstein's Topics in Algebra. I'm having trouble with the problem (working through the text independently). For $F$ a field, define $V_n=\{p(x)\in F(x) : \deg p(x)<n, n\in ...
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19 views

Ker and Im sum of matrix [on hold]

Suppose we have matrix and we have found Im and Ker as vectors.How to find Im+Ker?
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21 views

Name for problems where the constraints are on inner products

I have a problem with a lot of dot-product constraints like $V_1 \cdot V_2 = 0$ or $V_1 \cdot V_3 = V_2 \cdot V_4$. However, I don't know what these types of problems are called so I can't look up ...
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1answer
40 views

Projective Geometry in $\mathbb{R}^{3}$: “Lonely lines” in source/image planes

I am reading some lecture slides about projective geometry in $\mathbb{R}^{3}$. In particular, given a source plane, $S$, an image plane, $I$, and a focal point, $f$, the issue at hand is the ...