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 ...

learn more… | top users | synonyms

0
votes
1answer
76 views

Prove that when AC matrices are multiplied and equal the identity matrix the solution of Ax=b is consistent for all real numbers [duplicate]

Continuing on my newbie dive into linear algebra I have this problem: Suppose that $A = [a_{ij}]$ $m×n$ and $C = [c_{ij}]$ $n×m$ and $AC = I$ . Prove that the system Ax = b is consistent for every ...
2
votes
1answer
304 views

System of Linear Equations ($3\times 6$ matrix, parametric answer)

Solve the system \begin{array}{r@{}r@{}r@{}r@{}r@{}r@{}r@{}r} x_1 & - 2 x_2 & - 2 x_3 & & + 5 x_5 & - 4 x_6 & = & -1 \cr & & & - ...
2
votes
2answers
194 views

Show that when $BA = I$, the solution of $Ax=b$ is unique

I'm just getting back into having to do linear algebra and I am having some trouble with some elementary questions, any help is much appreciated. Suppose that $A = [a_{ij}]$ is an $m\times n$ matrix ...
0
votes
1answer
109 views

A problem on Vector space over $\mathbb{C}$ and $\mathbb{R}$

A text book problem asks if $\Bbb{C}$ is the field of complex numbers, which vectors in $\mathbb{C^3}$ are linear combinations of (1,0,-1) , (0,1,1) and (1,1,1)? Can it be safely assumed that is ...
0
votes
1answer
24 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 ...
0
votes
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 ...
7
votes
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
91 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?
1
vote
3answers
101 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 ...
0
votes
1answer
27 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
55 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 } & ... ...
1
vote
1answer
71 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 $[X]_{A,A}$ = ...
1
vote
0answers
65 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
79 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
votes
1answer
115 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
votes
4answers
723 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
votes
1answer
56 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 ...
1
vote
2answers
55 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 ...
1
vote
3answers
47 views

What is the equation for $y$ alternating between $0$ and $5$? [closed]

What is the equation for below data: Data: if $x = 0$ then $y = 5$. if $x = 1$ then $y = 0$ if $x = 2$ then $y = 5$ if $x = 3$ then $y = 0$ if $x = 4$ then $y = 5$ if $x = 5$ then $y = 0$ Can ...
4
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
votes
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
votes
1answer
165 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
213 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
93 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
104 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
votes
1answer
242 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 ...
0
votes
2answers
40 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 ...
0
votes
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
votes
1answer
950 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 ...
1
vote
2answers
71 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 $x\cdot y=0$?
0
votes
2answers
79 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
votes
1answer
293 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
123 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 $\det(I+A)>0$?
0
votes
2answers
90 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
174 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
116 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
177 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
271 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
168 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 ...
3
votes
1answer
6k 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
40 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 ...
0
votes
1answer
565 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.
16
votes
6answers
10k 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 ...
1
vote
3answers
1k 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
64 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: ...
2
votes
0answers
125 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 ...
1
vote
3answers
255 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
242 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 ...
1
vote
2answers
283 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
votes
2answers
972 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 ...