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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1answer
25 views

Choosing an ordered basis?

Let $X = Y = \mathbb{R}^{3\times2}$. Let $T : X \to Y$ be the linear transformation given by $T(S) = RS$ where $$R = \begin{pmatrix} 4 & 10 & 0\\ 0 & 2 & -2\\ 4 & 2 & 8 ...
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2answers
60 views

Is $\operatorname{span}\{(0,1)\}$ a point in $\mathbb{R}^2$?

Is $S_1 = \operatorname{span}\{(0,1)\}$ a point in $\mathbb{R}^2$? Does $0$ vector belong to $S_1$? Is $S_2=\operatorname{span} \{(0,1),(0,2)\}$ a line in $\mathbb{R}^2$? Does $0$ vector belong to ...
9
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4answers
2k views

Proof of elementary row operations for matrices?

I'm taking a Linear Algebra course, and we just started talking about matrices. So we were introduced to the elementary row operations for matrices which say that we can do the following: ...
2
votes
1answer
120 views

Pushouts and Pullbacks in Category Theory

How would one prove existence of pushouts and pullbacks where the objects are vector spaces and the morphisms are linear transformations?
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3answers
109 views

A problem about dual basis

I've got a problem: Let $V$ be the vector space of all functions from a set $S$ to a field $F$: $(f+g)(x) = f(x) + g(x)\\ (\lambda f)(x) = \lambda f(x)$ Let $W$ be any $n$-dimensional subspace of ...
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1answer
54 views

Finding out the range and statistical distribution

The range of the heights of the female students in a certain class is 13.2 inches, and the range of the heights of the male students in the class is 15.4 inches. Which of the following statements ...
2
votes
1answer
61 views

On invertibility of a special matrix - Hilbert matrix [duplicate]

I want to know how to prove that the below matrix is invertible \begin{pmatrix} 1 & \frac { 1 }{ 2 } & ... & \frac { 1 }{ n } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 3 } & ... ...
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1answer
74 views

Question about linear transformation proof

Let $V$ be a finite dimensional vector space over field $\mathbb{F}$ and $X, Y$ linear transformations from $V \mapsto V$. When do there exist ordered bases $A$ and $B$ for $V$ such that $[X]_{A,A}$ = ...
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0answers
79 views

Prove that the mean of all vectors is a centroid

I am looking for a nudge in the right direction: I am trying to prove that a vector is the mean for all vectors in some d-dimensional space is the centroid of the space. The distance between vectors ...
2
votes
1answer
82 views

A problem on linear transformation and invertibility

If $T$ is a linear transformation of rank one on a finite dimensional vector space, I have to check whether the statement "$I-T$ is invertible" is true or not ? Now suppose $I-T$ is invertible then ...
3
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1answer
165 views

Construct a basis for $\mathbb{R}^4$ given two vectors and any two of the standard basis vectors in $\mathbb{R}^4$

I think I came up with a solution, but I wonder if there's a "better" way to do it that my prof may be looking for. Construct a basis for $\mathbb{R}^4$ that consists of the vectors ...
2
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4answers
776 views

Matrix Algebra Question (Linear Algebra)

Find all values of $a$ such that $A^3 = 2A$, where $$A = \begin{bmatrix} -2 & 2 \\ -1 & a \end{bmatrix}.$$ The matrix I got for $A^3$ at the end didn't match up, but I probably made a ...
0
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1answer
68 views

Two linear maps which commute

If $S$ and $T$ are linear maps over a finite dimensional complex vector space. They commute i.e $ST=TS$. Is there any common subspace under which both are invariant ? I guess it is ...
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2answers
61 views

Finding the amount of solutions in a 3 equation solution

So, I'm not really sure how to calculate the amount of solutions for a system with 3 equations. All I know is that it has something to do with matrices and the discriminant, but I'm not sure where to ...
3
votes
3answers
116 views

How to prove that there does not exist a natural number '$n$' whose product of digits is $n^3-25n^2+151n$.

How to prove that there does not exist a natural number '$n$' whose product of digits is $n^3-25n^2+151n$. I don't know where to start. NOTE: I do not want the answer a hint should do it. Any help ...
2
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1answer
49 views

Why $V$ has has an orthonormal eigenbasis$\{e_i\}$ and $AA^*e_i=\lambda_i^2e_i$, where $\lambda_i\geq0$

For any $A\in M_{n\times n}(\Bbb{C}) $, why linear space $V$ has an orthonormal eigenbasis$\{e_i\}$ and $AA^*e_i=\lambda_i^2e_i$, where $\lambda_i\geq0$?
4
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1answer
204 views

A basic question on linear maps and upper triangular form

Let $S$ and $T$ be two linear maps from $V$ to $V$ ($V$ complex vector space) such that $ST=TS$. I need to prove that there exists a basis with respect to which both the matrices are in ...
2
votes
2answers
217 views

How find the possible eigenvalue of $P$

Question 1: Let $P$ be a real matrix such that $$P^{T}=P^2$$ What are the possible eigenvalues of $P$? I consider sometimes, But I can't, and I guess $1$ and $-1$? Thank you Question 2: if $P$ is ...
0
votes
1answer
95 views

Orthogonal Projection of y on range (x)

I've been working on a problem set for econometrics, I wanted to verify that I correctly understand what I'm doing… Given $x=\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $y=\begin{pmatrix} 1 \\ ...
1
vote
1answer
123 views

Abstract vector spaces linear transformations

Let V be a finite dimensional vector space over field F and X, Y linear transformations from V to V. When do there exist ordered bases A and B for V such that $[X]_{A,A}$ = $[Y]_{B,B}$? Prove such ...
3
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1answer
265 views

Vector space proof

Let $\mathbb F$ be a field and let $V$ be a vector spaces over $\mathbb F$. Show that for all $w\in V$, $(-1_{\mathbb F}) \cdot w=-w$, where $1_{\mathbb F}$ is the multiplicative identity of $\mathbb ...
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votes
2answers
45 views

linear dependence and independence

Let $u_1$,$u_2$,$u_3$,$u_4$ be vectors in $R^2$ and $$u=\sum t_ju_j, 1\le j\le4 $$:$t_j>0$ and $$\sum_{j=1}^4 t_j=1$$. Then Three vectors $v_1,v_2,v_3 \in R^2$ may be chosen from ...
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2answers
82 views

Find subspaces $W, X, Y \subset \Bbb{R} ^2$ with $\Bbb R ^2 = X \oplus Y$ but $X ∩ W = Y ∩ W = \{0\}$

Find subspaces $W, X, Y \subset \Bbb{R} ^2$ with $\Bbb R ^2 = X \oplus Y$ but $X ∩ W = Y ∩ W = \{0\}$. I don't know how to relate the direct sum to the intersection, does the cancellation theorem ...
0
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1answer
1k views

Is there a simple way to find a matrix whose null space is the span of a given set of vectors?

The problem, and my solution is outlined below. I think my solution is correct, but I feel as if I went about my solution in an awkward way, and that there may be a better/cleaner way to solve the ...
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vote
2answers
75 views

Problem in n dimensional space

If $x$, $y$ in $R^n$ are such that $\lvert x+ty\rvert \geq \lvert x\rvert$ for all $t \in R$, then how do I show that $x\cdot y=0$?
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2answers
101 views

Linear forms and the dimension of the intersection of their kernel

Let $E$ a finite dimensional vector space over a field $\mathbb F$ and $\dim E=n$ and let $(\ell_1,\ldots,\ell_k)$ a linearly independant family in $E^*$. The question is to prove that ...
0
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1answer
397 views

Finding coordinate vector relative to different bases

I have been working on coordinate vectors, change of basis, and matrix of linear transformation. However, I don't know how to find the coordinate vector with respect to different bases. Any help would ...
2
votes
2answers
133 views

is it true that $\det(I+A)>0$ , if $\det(A)>0$?

I saw an inequality for $n\times n$ matrices. I was wondering if the inequality is true or not? Does $\det(A)>0$ imply $\det(I+A)>0$?
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2answers
92 views

Eigenvalue: How to make it more “eigen”?

Suppose that $V$ is finite-dimensional linear space over $F$, $\lambda_1 ... \lambda_m \in F$, $E_1, ..., E_m$ are subspaces of $V$. The question is, what additional constraints can be given such ...
2
votes
1answer
228 views

Eigenvalues of a matrix $A$ and corresponding linear map (Linear algebra: Hoffman kunze 6.2.15)

Let $V$ be the vector space of $n\times n$ matrices over the field $F$. Let $A$ be a fixed $n\times n$ matrix over $F$. let $T$ be a linear operator on $V$ defined as $T(B) = AB$. Question is to ...
2
votes
1answer
165 views

Problem with sum of projections

Let $X$ be a real linear space, $(P_i)_{i=1}^n$ -a finite sequence of linear mappings $P_i :X\rightarrow X$ such that $P_i^2=P_i$ for $i=1,...,n$, $(P_1+...+P_n)^2=P_1+...+P_n$. I wish to show ...
1
vote
1answer
276 views

Using absolute coordinates in 2D affine transformation matrix

In my 2D animation program I have a sprite which transformation is described by a 2D affine transformation matrix (SVGMatrix): $$ \begin{bmatrix} a & c & e \\ b & ...
5
votes
4answers
355 views

if $AB\neq 0$ for any non zero matrix $B$ then $A$ is invertible

Question is to check that : If $A$ is an $n\times n$ matrix over a field $F$ and $AB\neq 0$ for any non zero matrix $B_{n\times n}$ over $F$ then, $A$ is invertible. This does make some sense to me ...
1
vote
1answer
259 views

Computing the number of positive and negative eigenvalues

Given a $n \times n$ symmetric matrix $A$ with integers as entries I would like to compute the number of strictly negative $\rm{nn}(A)$ and positive $\rm{np}(A)$ eigenvalues of $A.$ My question is ...
6
votes
2answers
10k views

Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar. I ...
1
vote
1answer
41 views

Why $A$ and $B$ are nilpotent if $A+\lambda_i B $ is nilpotent?

Let $A$ and $B$ in $\mathcal M_n(\mathbb C)$ and assume there are $n+1$ different complex numbers $\lambda_1,\ldots,\lambda_{n+1}$ such that for all $i$, $A+\lambda_i B$ is nilpotent. How prove that ...
2
votes
1answer
926 views

How to find whether the line is inside the polygon or outside.

I have a polygon How can i prove whether the black color line lies outside the polygon or inside the polygon . Given the coordinates of the black line and all the vertices of the polygon.
23
votes
6answers
19k views

Show that the determinant of $A$ is equal to the product of its eigenvalues.

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic ...
7
votes
5answers
3k views

Intuition behind Matrix being invertible iff determinant is non-zero

I have been wondering about this question since I was in school. How can one number tell so much about the whole matrix being invertible or not? I know the proof of this statement now. But I would ...
2
votes
1answer
65 views

How to find $x + y + z$?

Q. If $x^{1/3} + y^{1/3} + z^{1/3} = 0$, then (A) $x + y + z = 3 xyz$ (B) $x + y + z = 0 $ (C) $( x + y + z)^3= 27 xyz$ (D)$ x^3 + y^3 + z^3 = 0$ What I've done: ...
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0answers
160 views

How to justify my solution of this problem, even though I got the right answer?!

I have a problem that I would like to justify the solution of, even though I somehow got the right answer?! Problem: Let $u=(2,1,1)^t, v=(1,0,1)^t, u'=(1,1,0)^t, v'=(6,3,3)^t$, where $u,v,u',v'$ are ...
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3answers
301 views

Linear Algebra Problem - Ph.D exam

I stole this problem from a Ph.D exam from another university. Let $V$ be a real vector space and let $T: V \to \mathbb{R}$ be a linear transformation. Suppose $(v_1, \dots, v_n)$ is a bssis for ...
3
votes
4answers
286 views

minimal polynomial of a matrix with some unknown entries

Question is to prove that : characteristic and minimal polynomial of $ \left( \begin{array}{cccc} 0 & 0 & c \\ 1 & 0 & b \\ 0 & 1 & a \end{array} \right) $ is ...
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2answers
340 views

Proper subspaces of $ R^n $

I'm trying to prove that proper subspaces of $ R^n $ are closed and have empty interior. To prove that they are closed I'm trying to use the fact that invers images of closed sets are closed sets by ...
0
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2answers
1k views

Invertibility and Rank of matix

Can anyone give me a proof for, B is an invertible $n$x$n$ matrix, then the rank of $AB$ is the same as the rank of $A$ for every $m$x$n$ matrix $A$. Also, is the converse true for the statement ...
0
votes
1answer
83 views

Showing a given map is diagonalizable without calculating the eigenvalues

Let $f:\mathbb R^3\rightarrow \mathbb R^3$ be a linear map with matrix in the canonical bases is given by $$A=\left[ \begin {array}{ccc} 2&0&1\\ 0&2&-1 \\ 1&-1&1\end {array} ...
5
votes
2answers
460 views

Show that if $A^{n}=I$ then $A$ is diagonalizable.

Suppose $A$ is an $m \times m$ matrix which satisfies $A^{n}=1$ for some $n$, then why is $A$ necessarily diagonalizable. Not sure if this is helpful, but here's my thinking so far: We know that $A$ ...
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1answer
3k views

Finding determinant by applying Gaussian Elimination

(I don't know how to make a matrix here, someone please correct it into a better format, thanks~) So I'm applying the Gaussian Elimination to find the determinant for this matrix: $\begin{pmatrix} ...
0
votes
3answers
1k views

number of students in a class

A third-grade teacher has $n$ boxes, each containing 12 pencils. After the teacher gives $p$ pencils to each student in the class, the teacher has $t$ pencils left over. Which of the following ...
3
votes
1answer
177 views

A wrong proof that the kernel and image are always complementary

Let $E$ be a vector space, $f\colon E\rightarrow E$ an endomorphism. Let $A=\ker(f)\oplus \operatorname{im}(f)$; that is $$A=\{x\in E\;|\; \text{there exists a unique}\; (a,b)\in ...