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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13
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1answer
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Finding the basis of a null space

I am trying to understand why the method used in my linear algebra textbook to find the basis of the null space works. The textbook is 'Elementary Linear Algebra' by Anton. According to the textbook, ...
0
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1answer
18 views

How to shift a square into the direction of a given angle so that the old and new borders overlap

Recently I had a problem where I had to shift a square called r into a specific direction, e.g. $45$°. This is what you have: The center of ...
1
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0answers
53 views

Polar System with Short Answers, How $U(0, \theta)=\pi$ will be calculated?

I read some notes on Laplace. I ran into a short answer question as follows. We have a Laplace equation in Polar Systems: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial ...
0
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1answer
24 views

Isomorphism is an equivalence relation on finite dimensional vector spaces over $F$.

Show that isomorphism is an equivalence relation on finite dimensional vector spaces over $F$. A relation $R$ is an equivalence relation if it is: reflexive, i.e. $xRx$ for all $x$ symmetric, ...
3
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2answers
48 views

Understanding Eigenvalues, Eigenfunctions and Eigenstates

Please could somebody explain the meaning and uses of Eigenvalues, eigenfunctions and eigenstates for me. I have taken 3 years of physics and math classes at university and never fully grasped the ...
0
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1answer
25 views

Elements fail to form a basis

Consider the vector space $P$2 and the set $$5−1t+4t^2,−4+3t+1t^2,8+5t+kt^2$$ For which $k \in \mathbb{R}$, do these three elements fail to be a basis of $P$2? I thought in order to make the three ...
1
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1answer
36 views

Why does the additive inverse not follow

I need to prove that the vector space of $\mathbb{R}^2$ with the following operations: $x + y = (x_1 + 2y_1, 3x_2 - y_2)$ The usual scalar multiplication of $cx = (cx_1, cx_2)$ The answers in my ...
0
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1answer
24 views

Compute the matrix with the standard basis.

Let $X$ be the left representation of $S_3$. Compute the matrix $X(132)$ in the standard basis $S = {\{123,132,213,231,312,321}\}$. attempt: In general if $G = S_3 = {\{\pi_1, \pi_2,..,\pi_6}\}$, ...
2
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1answer
27 views

Proof of Cauchy-Schwarz Inequality 1

In my lecture notes I've written the proof of Cauchy-Schwarz inequality as: Let t $\in$ R and $\langle x+ty, x+ty\rangle \geq 0$, then $\langle x+ty, x+ty\rangle $ = $\langle x, x+ty \rangle + ...
1
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0answers
27 views

Eigenvalues of the subtraction of a gram matrix and a psd rank $1$ matrix.

If $V$ is $p \times r$ matrix ($r<p$) with rank $r$, then $G=V^TV$ is a $r \times r$ gram matrix with rank $r$. Let $x$ be a $r \times 1$ vector, and $xx^T$ is a psd rank $1$ matrix. It's already ...
0
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1answer
24 views

Proving left inverse for $A$

Taken from Artin's book, need to prove that an $m \times n$ matrix $A$ (where $m < n$) has no left inverse. A hint given is to compare matrix $A$ to an $n \times n$ matrix obtained by taking $A$ ...
0
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0answers
22 views

If two linear systems are equivalent, they have the same size augmented matrix. [on hold]

If two linear systems are equivalent, they have the same size augmented matrix? It is false but do any one know why for this?
0
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0answers
13 views

What star rating is representative of this distribution? [on hold]

100 people vote. They can vote 1, 2, 3, or 4 stars. Distribution: 1 = 33, 2 = 26, 3 = 12, and 4 = 28. What star rating would you say is "representative" of these 100 people: 2.36 (2), the average, ...
0
votes
0answers
12 views

Picture of Orthogonal Projection and Parallel Projection which is not a orthogonal projection in $3$ dimensions.

I would like to see a picture of an orthogonal projection in $3$ dimensions. I would also like to see a picture of a parallel projection that is not an orthogonal projection in $3$-dimensions. Does ...
1
vote
2answers
28 views

Understanding basis algorithm result

I've a matrix ${\bf A}$ defined as A = \begin{pmatrix} 1 & -2 & 0 & 3 & 7\\ 2 & 1 & -3 & 1 & 1\\ \end{pmatrix} And ${W_1}$ is the solution ...
1
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0answers
10 views

Eigenvalue ratio evolution of Laplacian matrix when add edges

Consider an connected digraph, we use the classic definition of the Laplacian matrix $L$: $L=D-A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. There has been many researches on ...
0
votes
1answer
17 views

Show that for each $v \in V$ exist $w \in W$ and $c \in \Bbb R$ unique such that $v=cv_0+v$

Let be $V$ a vector space over the field of real numbers, $f \in V^*$ and $W=ker (f)$. If $v_0 \in V$ is a vector such that $f(v_0)\neq0$, show that for each $v \in V$ exist $w \in W$ and $c \in \Bbb ...
2
votes
0answers
38 views

Supremum equal Max

Let $p$ be a polynomial and $\|.\|_A$ is a norm defined by $$\|\mathbf{x}\|_A:=\sqrt{\mathbf{x}.A\mathbf{x}},$$ for $\mathbf{x}\in\mathbb{R}^n$ and $A\in\mathbb{R}^{n\times n}$. Let $A$ be a symmetric ...
-2
votes
3answers
41 views

How do I determine if A is diagonalizable [on hold]

$A= \begin{bmatrix} -5 & 1 & 5 \\ -7 & 4 & 4 \\ -1 & 1 & 1 \end{bmatrix}$ How do I show that A is diagonalizable?
1
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2answers
29 views

Find a basis for Each corresponding eigenspaces

$A$= $\begin{bmatrix} -5 & 1 & 5 \\ -7 & 4 & 4 \\ -1 & 1 & 1 \end{bmatrix}$ I now want to find the eigenctvectors of $A$ and the basis corresponding to each eigenspaces. ...
0
votes
2answers
31 views

How to find the limit of a matrix $P^n = UD^nU^{-1}$ where $D$ is a diagonal matrix of eigenvalues and $U$ a matrix of eigenvectors?

If we have a matrix where $P = UDU^{-1}$, where $D$ is a diagonal matrix of real eigenvalues that are less than or equal to 1, and $U$ is the corresponding matrix of eigenvectors, how can we show that ...
12
votes
1answer
909 views

Eigenvalues and eigenvectors of Hadamard product of two positive definite matrices

The component-wise product (Hadamard product) of two positive definite matrices is a positive definite matrix (Schur product theorem). I encountered the following proof of it: $A=(a_{ij})$ and ...
1
vote
2answers
32 views

Matrices and diagonalization.

I could verify that $P$ statement is false by just calculating the determinant but couldn't answer $Q$ statement. Any clue about $Q$??
1
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1answer
14 views

Row Equivalent Matrices

If I have a matrix $A$, where there are zeros everywhere apart from the first row, what are the matrices that are not row equivalent to $A$. I know that if two matrices are row equivalent, we can ...
1
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1answer
50 views

Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
2
votes
2answers
38 views

A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
2
votes
2answers
2k views

Is the set of all invertible $n \times n$ matrices a vector space?

I'm studying Algebra and I'm asked to prove or disprove "Is the set of all invertible $n \times n$ matrices a vector space?" I assume with respect to the usual matrix-sum and scalar multiplication. I ...
0
votes
1answer
31 views

Finding a Diagonal Matrix for a Linear Transformation

here is the problem: I am pretty stuck on this one. I thought that the formula for a projection was: wx/ww times w, which in turn forms a matrix [w1^2, w1w2], [w1w1, w2^2] * 1/ (w1^2 + w2^2), but ...
0
votes
3answers
36 views

find when matrix is not diagonalizable

Let $A = \begin{pmatrix} 3&0&0\\ 0&a&a-2\\ 0&-2&0 \end{pmatrix}$ A is not diagonalizable find $a$. how can I tell when $a$ is diagonalizable by it's characteristic ...
1
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3answers
248 views

If $M, N$ are finite dimensional vector spaces with same dimension, then if $M$ is subset of $N$, then $M=N$.

If $M, N$ are finite dimensional vector spaces with same dimension, then if $M$ is subset of $N$, then $M=N$. I think i need to show that both vector spaces are spanned by the same bases in order to ...
2
votes
1answer
24 views

Finite abelian group as Z-module

If $M$ is a finite abelian group then $M$ is naturally a $Z$-module. Can this action be extended to make $M$ into a $Q$-module ?
-1
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1answer
37 views

Image of a linear transformation

Let $T : V \to W$ be a linear transformation. If $A$ is a subspace of $V$, show that its image, $$ T(A) = \left\{ T(x) \in W \mid x \in A \right\}, $$ is a subspace of $W$. I have no idea how ...
0
votes
2answers
33 views

It is true that $rank(xy^T)=1$? [on hold]

Let $x,y\in \mathbb{C}^n$. It is true that $rank(xy^T)=1$?
0
votes
3answers
102 views

When $\operatorname{im}(A) = \ker(A)$

Consider the following true/ false qustion: There exists a $2 \times 2$ matrix $A$ such that $\operatorname{im}(A) = \ker(A)$. I know that this is true, but I am not sure how to show it. If $A$ ...
4
votes
2answers
54 views

Testing the diagonalizability of matrix $B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$

How to show that the matrix $$B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$$ is diagonalizable when $a\neq0$, when ...
1
vote
1answer
31 views

Find $U\cap V$.

Find $U\cap V$. Given $$U = \text{span}{(1,1,-1),(2,3,-1),(3,1,-5)}$$ $$ V=\text{span} {(1,1,-3),(3,-2,-8),(2,1,-3)}$$ $A. U$ $B. V$ $C. \{0\}$ $D.$ None of the above ATTEMPT: I have found that ...
1
vote
4answers
70 views

How Many Times The Clock Hands Make an angle Theta?

You might have encountered such a question where $\theta=90^o$ but this a little different and is causing a little problem. I approached this problem in the following matter and solved for $\theta= ...
0
votes
1answer
28 views

Linear trasnformation kernel and image [on hold]

$V$ is a vector space. Let $T: V \to V$ be a linear transformation. Prove that if $\text{Ker}\: T = \text{Ker}\: T^2$ then $\text{Im}\:T = \text{Im}\:T^2$. How do I prove it?
0
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0answers
13 views

Piecewise linear function given three points and two crossover boundaries

Suppose you have three points; $(3500, 700)$, $(52500, 5075)$, and $(527500, 36800)$. As well as two $x$ boundaries $25000$ and $200000$. The question is then to construct three lines (each of which ...
1
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1answer
28 views

Anticommuting matrices and their eigenvalues

Let $A,B\in \mathcal{M}_n(\mathbb{C})$. It is known that if $AB=BA$ and $\lambda_1, \lambda_2, \dots, \lambda_n $ are the eigenvalues of $A$ and $\beta_1, \beta_2, \dots, \beta_n$ are the ...
3
votes
1answer
13 views

Linear independent set of functionals makes certain map surjective

Let $V$ be a finite $n$-dimensional vector space over a field $K$ and $\{\lambda_{1},\ldots, \lambda_{n}\}$ be a linearly independent set of functionals Show that the linear map $$\Lambda:V\to K^n$$ ...
0
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0answers
24 views

Matrix transformation into block off-diagonal form

Consider the 4-by-4 matrix $\boldsymbol M = \boldsymbol M_0 + \boldsymbol M_1$, where $\boldsymbol M_0 = \alpha \left( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 ...
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votes
1answer
22 views

To find dimension of $N(A) \cap R(B)$ over R [on hold]

To find dimension of $N(A) \cap R(B)$ over R A = $\begin{bmatrix} 1 & 2 & 0 \\ -1 & 5 & 2 \end{bmatrix}$ B=$\begin{bmatrix} 1 & 2 \\ -1 & 0 \\ 3 & 1 \end{bmatrix}$ i ...
0
votes
1answer
18 views

Let $(L_1,L_2,L_3)$ be an ordered triple of pairwise distinct plane in $K^3$. There is two possibles types of relative arrangement of such triples.

Let $(L_1,L_2,L_3)$ be an ordered triple of pairwise distinct plane in $K^3$. Prove that there is two possibles types of relative arrangement of such triples characterized by the fact that $\dim ...
0
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0answers
24 views

Proof that $\operatorname{End}(V) \rightarrow Gl_n(K), F \mapsto M_A^A(F)$ denotes a group-isomoprhism.

Definition: Let $A$ be a Basis of $V$, $V$ a $K$ - Vectorspace. $M_A^A(F) = \Phi_A \circ F \circ \Phi_A^{-1} $, where $\Phi_A$ denotes the following function: $n := \dim V, \{x_1,\ldots,x_n\} = A$ ...
1
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0answers
71 views

Spliting subspaces and fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
0
votes
1answer
30 views

Determinant of an elementary matrix

I read a very slick proof of determinant properties, in this case of the fact $\det A = \det A^T$, which says in one place It suffices to notice that for any elementary matrix $M$ we have $\det M ...
8
votes
7answers
9k views

Real life applications of general vector spaces

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of ...
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3answers
73 views

A is a square matrix and given that $A^3 = 2\mathbb{I}$, then show $A-\mathbb{I}$ is invertible and find its inverse [on hold]

Could anyone guide along with this question? I was trying $(A−I)(A−I)^{−1} = I$ and was figuring if there's a way out to expand $(A−I)^{−1}$. I also tried $(A−I)x=0$ but to no avail.
0
votes
0answers
19 views

Determining if a Polynomial is a subspace and its Basis

Hi, the question is Which of the subsets of P2 given in Exercises 1 through 5 are subspaces of P2 Find a basis for those that are subspaces. So I know that P'(1) = 1b + 2c And I know that P(2) = a ...