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 many solutions in n x n matrix?

Let $A$ be an $n \times n$ matrix, whose column vectors are linearly independent; How many solutions does the homogeneous linear system $A\mathbf{x}=0$ have? Could any one help to how to solve the ...
2
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1answer
20 views

Proving positive definite of a specific matrix.

If $A = D + L + L^T $, where D is the diagonal of $A$ and L is the lower triangular matrix, is symmetric and positive definite, then: $$ M_\omega= \left(\frac{1}{\omega}D + L\right) ...
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1answer
13 views

Pseudoinverse of $KXK$, $K=\left(\begin{array}{cc}1_{N-1}& 0 \\ 0 & 0\end{array} \right)$Pseudoinverse of $KXK$, K=

I am searching for $(KXK)^+ =\left(\begin{array}{cc}X_{N-1}& 0 \\ 0 & 0\end{array} \right)^+$, where $X^{-1}\in\mathbb{R}^N$ exists and $K=\left(\begin{array}{cc}1_{N-1}& 0 \\ 0 & ...
1
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1answer
31 views

Explanation about the shape of an object of a linear transformation.

We have the linear mapping $A: R^2 \to R^2$. In which case (see picture) we can conclude that $A$ might be a linear mapping? a.) and d.) are not correct because we can immediately see that the zero ...
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3answers
34 views

Why is this matrix invertible [duplicate]

I was wondering if there is a way to see why $(1+A)$ invertible, if $A$ is a skew symmetric matrix. and I know that all eigenvalues of $A$ have zero real part and $A$ is unitarily diagonalisable.
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0answers
18 views

Degenerate subspace

A null vector is a nonzero vector that is orthogonal to itself. If W is a subspace of V,let $W^{\perp}$ = [$v{\in}$ W : $v{\perp}$W]. $W^{\perp}$ is a subspace of V called W perp. A subspace W of ...
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votes
3answers
63 views

What are the possible eigenvalues of a linear transformation $T$ satifying $T = T^2$ [duplicate]

Let $T$ be a linear transformation $T$ such that $T\colon V \to V$. Also, let $T = T^2$. What are the possible eigenvalues of $T$? I am not sure if the answer is only $1$, or $0$ and $1$. It holds ...
1
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1answer
31 views

Relating the singular values of a scaled matrix to its determinant

Let $A$ be a real, $n\times n$,full rank matrix with singular values: $\sigma_1\ge\dots \ge \sigma_n$. Assuming the rows of $A$, $a_1,\dots,a_n$ are scaled so that $\|a_i\|_2 = 1$ for $i=1,\dots,n$, ...
3
votes
2answers
54 views

Find all complex matrices $A$ such that $n\operatorname{Tr}(AB) = \operatorname{Tr}(A)\operatorname{Tr}(B)$ for all $B$. [duplicate]

Consider a bilinear form $f(A,B) = n\operatorname{Tr}(AB) - \operatorname{Tr}(A)\operatorname{Tr}(B)$ defined on $M_n(\mathbb{C})$. I need to find the set $U^\perp$ of all matrices $A$ such that ...
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2answers
41 views

a problem from algebra of matrix [on hold]

Let $A=(A_{ij})$ be a matrix of order $n$. Where $A_{ij}=i^2+j^2$. Find rank of $A$.
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2answers
38 views

What does the following statement means?

I am reading these slides.. http://amath.colorado.edu/faculty/martinss/2014_CBMS/Lectures/lecture05.pdf But I am not able to understand the following: What is so special about orthonormal matrix.. ...
0
votes
2answers
35 views

Showing there is a projection between a normed space and a subspace

Problem: Let $E$ be a normed space. Suppose $A$ is a finite dimensional subspace of $E$. Show that there exists a continuous projection $T: E \to A.$ Proof. I can write $E=A\oplus B$, where $B$'s ...
0
votes
1answer
19 views

Linear Span of R3

I am stuck with this question from my assignment in which its given that W1 = L{(1,1,0),(-1,0,2)} and W2 = L{(1,0,2),(-1,0,4)} and it being asked to show that W1 + W2 = R3. Following are my ...
2
votes
5answers
82 views

Do planes stop, or are they ever expanding?

I am trying to understand sub-spaces in linear algebra and one of the rules mentions if W is my subspace then if k is any scalar and u is any vector in W then ku is in W. I am unsure how this works ? ...
0
votes
1answer
45 views

How to prove a linear transformation is onto if it is one to one. [on hold]

How do I prove that the generic linear transformation $ T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is onto if I only know that it is also one to one?
2
votes
1answer
48 views

Show that the given transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$ is linear by showing it is a matrix transformation

$R$ rotates a vector $45^{\circ}$ counterclockwise about the origin. So, I have a vector and its tail part begins at the origin. It is $45^{\circ}$ and will rotate to the left. I don't have any ...
4
votes
2answers
99 views

Prove that for any positive integer $n$, $A^n ≠ I$.

Let $A$ be a $2\times 2$ matrix with $tr(A) > 2$. Prove that for any positive integer $n$, $A^n ≠ I$. I feel like I should approach this with respect to eigenvalues, i.e. the sum of the ...
0
votes
1answer
16 views

Let $\dim(V)<\infty$ . Do there exist endomorphisms of $V$ satisfying the condition that $σ_1 + αβ − βα$ is nilpotent?

Let $V$ be a vector space finitely-generated over $\mathbb{C}$. Do there exist endomorphisms $α$ and $β$ of $V$ satisfying the condition that $σ_1 + αβ − βα$ is nilpotent? I just know that if ...
2
votes
3answers
70 views

Matrix Problem of form Ax=B

The matrix $A$ is given by $$\left(\begin{array}{ccc} 1 & 2 & 3 & 4\\ 3 & 8 & 11 & 8\\ 1 & 3 & 4 & \lambda\\ \lambda & 5 & 7 & 6\end{array} \right)$$ ...
1
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2answers
45 views

Let $V$ be a vector space and $\alpha$ a nilpotent endomorfism (of degree $k$), how can I show that $\alpha(x)+x$ is epic?

Let $V$ be a vector space and $\alpha$ a nilpotent endomorfism (of degree $k$), how can I show that $\alpha(x)+x$ is epic? If $v\in V$ I want to show that there exists $x\in V$ such that ...
1
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1answer
54 views

Dividing a Matrix into three parts

The matrix $A$ is given by $$\left(\begin{array}{ccc} 1 & 2 & 1 \\ 1 & 1 & 2 \\ 2 & 3 & 1 \end{array} \right)$$ Given that $A^3$ can be expressed as $A^3$=$aA^2+bA+cI$, find ...
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2answers
41 views

What will happen if we remove the hypothesis that $V$ is finite-dimensional in this problem

Original problem: Suppose $V$ is finite-dimensional and $S, T,U ∈L(V)$and $STU = I$. Show that T is invertible and that $inv(T) = US$. I know that it is because of the hypothesis of ...
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0answers
27 views

Pfaffian of skew symmetric block matrix

I am trying to prove that if A is a $m x m$ real matrix and $B=\begin{bmatrix}0 & A\\-A^t & 0\end{bmatrix}$, then Pfaffian(B)= $(-1)^{m(m-1)/2}$ det(A). I know that Pfaffian$(B)^2=$det(B), so ...
0
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0answers
20 views

How to calculate whether a ray intersects an arbitrarily oriented bounding box?

The bounding box is not axis-aligned. It can be rotated. I now have the origin of the ray, and direction and each coordinate of the eight vertices of the bounding box. what can I do to find out this ...
0
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0answers
25 views

When is system of linear equations smooth

I am wondering when is a system of linear equations smooth? More specifically, for Ax=B, what property of A guarantees smoothness of systems of linear equations? If it is known that A is always ...
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0answers
11 views

Interpretation of associativity of binary internal direct sums in vector spaces

Say $V,W,L$ are subspaces of a vector space $U$. I'm having some problems understanding what $V\oplus(W\oplus L)$ means. From what I understand this is equal to $V+W+L$ and it also says that every ...
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0answers
7 views

Ellipsoid confedence intervales?

Are Bonforonni, Scheffe, Multivariate t, and Tukey for simultaneous Confidence intervals are ellipsoid? How can I tell from the form of the interval that it is ellipsoid or rectangular?
0
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1answer
36 views

Does an orthogonal decomposition of a vector space exist?

Let V be a complex vector space equipped with an hermitian form (not necessarily positive definite), W a finite dimensional subspace of V such that it has zero radical (intersection between W and its ...
2
votes
3answers
27 views

Base of the $\mathbb{R}$ vector space that contains all real functions: $f(x) \not= 0$ for finitely many x $\in\mathbb{R}$

I did already prove that this is a vector space. It is easily shown that addition and scalar multiplication with functions that hold the above property again yields a function with $f(x) \not= 0$ for ...
0
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1answer
24 views

under change of coordinates the variety $Z(H_1,..,H_r)$ becomes $Z(x_1,…,x_r)\subset \mathbb{P}^n$.

A set $V\subset \mathbb{P}^n$ is called a linear subvariety of $\mathbb{P}^n$ if it's the zero locus of $r$ homogeneous and linear, i.e $V=Z(H_1,...,H_r )$ where each $H_i$ is a form of degree 1. I ...
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4answers
88 views

How to write the set of all permutations on a set $n=\{1, 2, \ldots, n\}$

Let $n ∈ N$. Let $S_n$ denote the set of permutations on $\{1, . . . , n\}$. For any $σ ∈ S_n$, define $sign(σ) := (−1)^N$ , where $σ$ can be written as the product of $N$ transpositions. Now, let ...
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votes
2answers
50 views

Why are the columns of a $3\times 5$ matrix linearly dependent?

If $A$ is a $3 \times 5$ matrix, explain why the columns of $A$ must be linearly dependent? The Rank Theorem tells me that $rank(A) + nullity(A) = n$ where $n$ represents the total number of ...
1
vote
1answer
24 views

Finding the dimension of the orthogonal complement

Let $U=M_{n}(\mathbb{C})$, and we define a bilinear form $\xi (A,B)=n\cdot tr(AB)-tr(A)tr(B)$. How do I find $dim(U_{\perp })$? I know that $U_{\perp }=\{A \in M_{n}(\mathbb{C})|\ \forall B\in ...
1
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1answer
19 views

Find Matrix $S$ such that $S'CS=\left(\begin{array}{cc} 1_{N-1} & 0 \\ 0 & 0 \end{array} \right)$ where $C:=1_N-\iota \iota'$

The Centering Matrix $C:=1_N-\iota \iota'$ has eigenvalue $1$ of multiplicity $n − 1$ and eigenvalue $0$ of multiplicity $1$. Therefore a matrix $S$ with columns consisting of eigenvectors of $C$ can ...
1
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2answers
17 views

linear independency in equation of linear span

we got the following vectors: $$v_1, v_2, w_1, w_3 \in V$$ $V$ is a vector space so that $\DeclareMathOperator{Sp}{Sp}\Sp\{v_1,v_2\} = \Sp\{w_1,w_2\}$ it's also defined that $\{v_1,w_2\}$ is linear ...
0
votes
1answer
15 views

Summation notation for the “ij'th” entry of matrix $(AB)^t$.

I'm just trying to figure out how to write out a formula to find the ij'th entry of the transpose of a matrix product. We have an $l \times m$ matrix $B$ and an $m \times n$ matrix $A$. We have $B = ...
0
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0answers
24 views

Eigenvectors of a real positive semi-definite submatrix

Let $\mathbf{A}$ be a real positive semi-definite matrix, and $\mathbf{V}$ and $\mathbf{\lambda}$ its eigenvectors and eigenvalues, respectively. I am wondering what is the relationship between ...
2
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0answers
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Notation for answer to basis of image (or kernel) problem?

For a certain transformation T, I was suppose to find the basis for the image and kernel for T. My question is how you would state the answer to such a problem. Is it okay to say that the basis for ...
2
votes
0answers
22 views

bilinear forms on $M_{n, n}(K)$

Let $K$ be a field and $V = M_{n, n}(K)$ the ring of $n \times n$ matrices over $K$. For any $f \in V^*$ (the dual space of $V$), we set: $\gamma_f: V \times V \to K, (A, B) \mapsto f(A B^t)$. I now ...
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1answer
23 views

Find the dimension the subspace

how can I find the dimension the subespace of R(x)_3 which is defined by (p(x)=ax^3+bx^2+cx+d;p(-1)=0)? Thanks in advance.
1
vote
1answer
30 views

Find the solution to the initial value problem $\overline{x}' = A\overline{x}, x(0) = \begin{bmatrix}{2} \\ {28}\end{bmatrix} $

$A = \begin{bmatrix}{16/3} && {1/3} \\ {-64/3} && {32/3}\end{bmatrix} $ I got the general solution $\overline{x} = c_1 e^{8t} \begin{bmatrix}{1} \\ {8}\end{bmatrix} + c_2 ...
2
votes
2answers
69 views

Each eigenvalue of $A$ is equal to $\pm 1$. Why is $A$ similar to $A^{-1}$? [duplicate]

$A$ is a non-singular matrix ($n \times n$) and each eigenvalue of $A$ is equal to $\pm 1$. Why is $A$ similar to $A^{-1}$? (by Jordan form)
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votes
1answer
26 views

In an augmented matrix representing a system of equations, why is it a contradiction when the LHS isn't zero and RHS is zero but not when flipped?

In an augmented matrix representing a system of equations, say a $1\times 3$ matrix: $(a,b \mid c)$, why is it a contradiction when $a=b=0, c\neq 0$ but not when $a,b\neq 0, c=0$ ?
3
votes
1answer
80 views

Power series as fractions

This is what I did: \begin{equation*} (x^3-x^6)x^6[x+x^2+x^3+..], \\ \frac{(x^3-x^6)x^6}{1-x}. \end{equation*} What mistake did I make? And, How to solve this: $1+3x^2+9x^4+27x^6+...+3^{157}x^{314}$ ...
2
votes
2answers
22 views

Determine how a linear mapping acts on a vector in general, by looking at how it acts on the basis.

My doubt is can this be done if the two vector spaces involved in the case are not isomorphic. We have the linear mapping $A: R^2 \to R^3$ and we know that $A(1,0) = (1,1,1)$ and $A(1,1) = (0,-1,1)$. ...
4
votes
2answers
306 views

A bad Cayley–Hamilton theorem proof

Given $A\in M_{n \times n}(\mathbb{F})$ and $p_{A}(x)=\det(xI-A)$ why saying that $\det(AI-A)=0$ is not valid?
0
votes
1answer
16 views

Nullspaces relation between components and overall matrix

If matrix $ C = \left[ {\begin{array}{c} A \\ B \ \end{array} } \right] $then how is N(C), the nullspace of C, related to N(A) and N(B)? The answer was that N(C) is the intersection of N(A) and ...
0
votes
1answer
18 views

Finding the the element-wise ratio of two column vectors

I have the following equation: $\mathbf{C} \mathbf{s_{+}} = \mathbf{s_{-}}$ where $\mathbf{C}$ is a $N \times N$ matrix of full rank, and $ \mathbf{s_{+}}$ and $ \mathbf{s_{-}}$ are column vectors. ...
0
votes
1answer
19 views

Diagonalizing a $3$x$3$ Matrix

Hello, everyone. I'm having difficulty with this homework problem. I solved for the eigenvalues and corresponding eigenvectors, but diagonal matrices can't have a zero in the diagonal. I'm not sure ...
0
votes
1answer
28 views

The necessary and sufficient condition for the three planes intersect in a straight line.

What's the necessary and sufficient condition for the three planes $a_ix+b_iy+c_iz=d_i,i=1,2,3$ intersect in a straight line.