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 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
282 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
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4answers
264 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
322 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 ...
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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 ...
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1answer
77 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} ...
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2answers
393 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
2k 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} ...
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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
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1answer
156 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 ...
2
votes
1answer
79 views

Finding a system of linear equations from solutions

Is there a simple way to find the system of linear equations given the solutions? For example, find a system with 2 equations and 3 variables that has solutions (1, 4, -1) and (2, 5, 2). I get: ...
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1answer
81 views

Sharp bound on off-diagonal entries of matrix with 1's on diagonal to make matrix invertible

Suppose $A$ is an $n \times n$ matrix with all 1 on the diagonal. What is the sharp bound $\epsilon(n)$ so that $A$ is invertible if all off-diagonal entries of $A$ have absolute value less than ...
2
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1answer
105 views

Homework - ERO with Unknown in Matrix

I was having a problem with how to properly perform elementary row operation (ERO) on a matrix. In the question, we were given an augmented matrix ...
4
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0answers
86 views

The determinant of a special matrix

Recently, I encounter the problem of calculating the determinant of the following matrix $$\left(\begin{array}{cccc} \sin(\theta_1) & \sin(\theta_1 + \delta_1) & \cdots & \sin(\theta_1 + ...
3
votes
3answers
238 views

Expressing the trace of $A^2$ by trace of $A$

Let $A$ be a a square matrix. Is it possible to express $\operatorname{trace}(A^2)$ by means of $\operatorname{trace}(A)$ ? or at least something close?
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2answers
1k views

Find a matrix so that $A^2$ not equal to 0 but $A^3$ is [Strang P78 2.4.23]

(a) Find a nonzero matrix $A$ for which $A^2 = 0$. (b) Find a matrix that has $A^2 \neq 0$ but $A^3 = 0$. Solution for (a): Let $A := \text{column $\times$ row} = \mathbf{cr^T} \neq \mathbf{0}$ ...
2
votes
1answer
212 views

How to solve this equation with multiple square root terms?

$(T^2-512)^{0.5}+(T^2-620)^{0.5}+(T^2-812)^{0.5}+(T^2-972)^{0.5}+(T^2-1100)^{ 0.5}=3T$ I first tried to just square each side to get rid of the square roots. But there were more square roots ...
2
votes
1answer
426 views

Rotation matrix convention; successive rotations in intermediate coordinate systems or not

I am getting very confused about the different conventions used for rotation matrices. Thing is I want to accomplish (3) successive rotations each time in the newly defined coordinate system. I use ...
0
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1answer
52 views

Differences of skew symmetric matrices

Let $A$ be an invertible real skew-symmetric matrix, and consider the difference $A_R:=RAR^{-1}-A$, for orthogonal $R$. Is it true that $A_R$ is either zero or invertible? Does the answer depend on ...
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2answers
110 views

Sum of projections onto vectors which sum to zero.

Premise: Let $V$ be some vector space over $K$ (in my case it's $d$ dimensional Euclidian space). Let $U= \{\mathbf u_i:i=1...n\}$ be a subset of $V$ which has the additional properties $\sum_i ...
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2answers
82 views

Determining which values a coefficient can take

I have the following system and matrix: $-x_1 + 3x_2-2x_3 = 7\\4x_1 + 2x_2-6x_3 = 14\\4x_1 + 5x_2+\textit{a}x_3 = 23\\$ $\begin{pmatrix} -1 & 3 & -2\\4 & 2 & -6\\4 & 5 & ...
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0answers
37 views

kernel of linear application

Let $n \in \mathbb{N}^*$, $A \in \mathbb{R}^{n}$, $A \neq 0$ and $\Phi_{A} \, : \, \mathcal{M}_{n}(\mathbb{R}) \, \rightarrow \, \mathbb{R}$ such that : $\forall M \in \mathcal{M}_{n}(\mathbb{R}), ...
0
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1answer
108 views

Composition of linear transformations

Prove that if two linear transformations of rank 1 $f,g$ have equal kernels and images, i.e. $\mbox{Ker}f=\mbox{Ker}g$, $\mbox{Im}f=\mbox{Im}g$ then $fg=gf$. Any help would be appreciated, I don't ...
2
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2answers
56 views

having trouble with recognizing the dimension

Hi how can I find the dimension of a vector space? For example : $V = \mathbb{C} , F = \mathbb{Q}$ what is the dimension of $V$ over $\mathbb{Q}$?
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1answer
358 views

ker(AB) = ker(A) + ker(B)

I'm trying to prove the following: Let $A$ and $B$ be two commutative square matrices ($AB=BA$) over a commutative field such that $Im(A)=ker(A)$ and $Im(B)=ker(B)$. Then $ker(AB) = ker(A) + ker(B)$. ...
0
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1answer
107 views

Finding a solution for ratio problem

I understand how to get the answer but can't we write explicitly the condition as a equation? I mean i have done this with length process, I need a quick process.
0
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1answer
83 views

Linearly Independent list of length 2

A list $S$ of length $2$ is linearly independent if and only if neither vector is a scalar multiple of the other. I'm not entirely sure precisely how to show this, but here are my thoughts: Let ...
1
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1answer
2k views

Homework: Which of these sets are closed under scalar multiplication?

One of the questions in an assignment I'm working on is as follows: Which of the following sets in $\mathbb{R}^4$ are closed under the usual operation in $\mathbb{R}^4$ of multiplication by ...
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2answers
88 views

Norms and Inner Products

Is it possible to form an inner product from a norm or only the other way around? When do inner products not have norms?
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1answer
39 views

Let $u=(1,0,1)$ and $X = \{w \in \Bbb{R}^3 | w \times u = 0\}$ Is this a subspace of $\Bbb R^3$?

Let $u=(1,0,1)$ and $$X = \{w \in \Bbb{R}^3 | w \times u = 0\}$$ Is X a subspace of $\Bbb R^3$? If so, find a spanning set for $X$ and give a complete geometric description. So I get that the ...
2
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0answers
166 views

Eigenvalues of differential operator

If $L : C^2[a,b] \rightarrow C^0[a,b] $ is s.t. $L y(t) = \ddot y(t) +p \dot y + q y(t) $ and $L$ is invertible then $L^{-1}$ has at most countable eigenvalues and they accumulate in $0$. Why ...
9
votes
2answers
136 views

Almost projections in matrix algebras.

I have a question about projection in matrix algebras over the complex number, that I can not solve.. Let p be a matrix in some $M_n(\mathbb C)$ and suppose that p is almost self-adjoint (i.e. ...
3
votes
2answers
605 views

What is the distance between the line and plane if it is parallel?

So far, I've gotten that the line is parallel to the plane $x = 2 + t$, $y = -3 + 2t$, $z = 1 + 4t$ With the vector of that being $U$ is $(1,2,4)$ and the plane $2y-z = 1$ with the vector $V$ being ...
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0answers
52 views

Solving a system of partial derivatives.

$uu_{x} - vv_{y} = 0$ $uu_{y} + vv_{x} = 0$ The subscripts represent partial derivatives. In general, the solution to this system should just be $0$. Not sure how to get that though. I was playing ...
4
votes
1answer
199 views

How do we find non-self-adjoint A and unitary U such that exp(iA) = U?

The following is a theorem: If $A$ is a self-adjoint matrix (i.e. $A^\dagger = A$), then $U = e^{iA}$ is a unitary matrix. This is easy to prove: $(e^{iA})^\dagger = e^{-iA^\dagger} = e^{-iA}$, ...
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votes
0answers
37 views

Is there an analogue/primitive of PCA which can be in a metric space rather than a vector space?

Principle component analysis PCA is done in a vector space, basically projecting a given vector onto the eigen vectors of the covariance matrix. I'd like to think of a primitive analogoue of PCA, ...
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3answers
134 views

Rating changes and probability calculations for chess world championship

I have some interesting questions that have to do with the rating changes and calculations for the Anand-Carlsen world championship. Most of this has to do with solving a system of linear equations. ...
0
votes
1answer
205 views

How to find the affine transformation(s) ( if any ) that maps one quadrilateral into another.

Given the Fundamental theorem of affine geometry. Let P,Q,R be any three non-collinear points in R2, and let U, V,W be any three other such points. Then there is exactly one affine transformation ...
5
votes
3answers
202 views

non linear transformation that satisfies $T(cx) = cT(x)$

I am just curious if there is a transformation that does not satisfy $\;T(x+y) = T(x) + T(y),\;$ but satisfies $\;T(cx)=cT(x).\;$ I cannot think of any. Thanks for any help people.
2
votes
2answers
98 views

Linear algebraic group of $GL(V)$ is independent of choice of basis

This is an easy question, but I am confused now. Please see the definition in this book In the last line. If we have $f(g)=0$, where $f(y)=p(x_{11}(y),...,x_{nn}(y))$, then after choosing different ...
2
votes
1answer
89 views

Homework: determining if a space is a subspace

This question is from an assignment I'm working on: Which of the following are subspaces of ${\bf F}~[-1,1]=\{~f~|~f:[-1,1]\rightarrow\mathbb{R}\}$? $X=\{~f\in{\bf F}[-1,1]~|~f(-1)=f(1)\}$ ...
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1answer
171 views

What is $\rho$ and $\sigma$ in this theorem?

This might be a silly question but, heres a note I made in linear algebra class: Suppose we have $Ax = \lambda x$, then $\rho(A)x = \rho(\lambda)x $, so $\rho(\sigma(A)) \subset \sigma(\rho(A))$. My ...
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1answer
38 views

How to prove $\|AB\|_F \le \|A\|_F\|B\|_2$?

$A,B$ are matrices (not necessarily square). $\|\cdot\|_F$ and $\|\cdot\|_2$ are the matrix Frobenius-norm and 2-norm, respectively. It suffices to show $\|AB\|_F^2 \le \|A\|_F^2\|B\|_2^2$. Let's ...
3
votes
2answers
133 views

Nilpotent matrix in $3$ dimensional vector space

This is a part of a long problem and I'm stuck in two questions of it. Let $E$ a $3$ dimensional $\mathbb R$- vector space and $g\in\mathcal{L}(E)$ such that $g^3=0$. So the first question is to ...
2
votes
3answers
196 views

How to measure “linear dependence” of more than two vectors?

I am looking for a way to measure the linear dependence of more than two vectors. For two vectors, we know that one way to measure its linear dependence is the angle between the two. If the vectors ...
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1answer
48 views

The DOF of $S^2$

As we know ,if $X_1,X_2,\cdots,X_n$ are samples ,the sample average is:$$\bar{X}=\frac{1}{n}\sum_{i=1}^nX_i$$ and the sample variance is $$S^2=\frac{1}{n-1}\sum_{i=1}^n(\bar{X}-X_i)^2$$ my textbook ...
3
votes
2answers
164 views

Mathematical Induction that seems v difficult

i have a problem with a mathematical induction but i find it really hard to solve: Q: $\sum_1^n iax^i = \frac{ax(1-x^{n}-nx^{n}+nx^{n+1})}{(1-x)^{2}}$ n is all positive integer I know it can be ...
0
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1answer
2k views

Minimal Spanning Set vs Basis of a vector space

I read the following in my textbook: Find as small a set of vectors that span the row space of $A$ as you can. Such a set is called a minimal spanning set. Is this terminology synonymous with ...
4
votes
5answers
2k views

What is an intuitive definition of the zero vector?

I'm learning now about vector spaces and subspaces, and one of three rules that determine if something is a subspace of a larger vector space is that it must contain the zero vector... but ...
2
votes
2answers
124 views

Proof that this is independent

Prove that {$1, \sin(x), \sin(2x), \sin(3x),\ldots, \sin(nx)$} is an independent set. I can think of the long way which is to differentiate this and put the differentiations into a matrix and row ...