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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Induction proof: B-spline property

Could any help me with the steps in the induction proof of the elementary B-spline property $$ \int_{-\infty}^{\infty} N_m(x) dx = 1 $$ For $N_1(x):=\chi_{0,1}(x)$, so the definition of convolution ...
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0answers
10 views

Solving LES containg spherical coordinates

i have a three-dimensional parametric equation of a line, where the directional vector is normalized and converted to spherical coordinates to calculate a angle offset. It looks like this: ...
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1answer
10 views

Orthogonal set proof?

Isn't this just the definition of an orthogonal set? What needs to be done to actually prove this?
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1answer
43 views

Prove that A(AB-BA) = (AB-BA)A implies AB-BA is nilpotent.

Let A and B be $n \times n$ complex matrices such that $A(AB-BA) = (AB-BA)A$ a) Show that for every positive integer $k$, the matrix $(AB-BA)^k$ is of the form $AC-CA$, where $C$ is an $n \times n$ ...
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1answer
13 views

Equation for adjoint transformation and proof.

I am really lost on this one. Any help would be appreciated. I'm very confused.
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8answers
233 views

To find eigenvalues

find the eigenvalues of the $6\times 6$ matrix $$\left[\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 ...
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1answer
21 views

Transformation self adjoint proof

Let $T$ be a linear operator on an inner product space $V$. Let $U_1 = T+T^*$ and $U_2 = TT^*$. Show that $U_1$, $U_2$ are both self-adjoint. I understand these just as innate properties. I don't ...
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1answer
20 views

Give an example of a singular matrix in $M_{3×3}(Q)$ the entries of which are distinct prime positive integers, or show that no such matrix can exist.

I know that the matrix exist because the entries are primes but I don´t know how to explain, i need some help. Give an example of a singular matrix in $M_{3×3}(Q)$ the entries of which are distinct ...
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0answers
27 views

Is $(A-A^{-1})$ skew-symmetric?

If $A$ is orthogonal, $(A-A^{-1})^T=A-A^T\neq -(A-A^{-1})=A^{-1}-A$ If $A$ is involutory, do we have an exception? In that case $(A-A^{-1})=0$, which seems trivial.
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1answer
23 views

Are symmetric matrices necessarily positive-definite / positive semi-definite?

I am trying to prove this just to be clear about this but I don't have enough conditions to force this idea to be true, so I doubt it is. Are symmetric matrices always at least positive ...
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0answers
4 views

Normalizers of subgroups of simple tensors

Inside $GL(2^n,\mathbb{C})$, we have subgroups that are formed by simple tensors. For example, in $GL(16,\mathbb{C})$, we've got subgroups like $H \leq G \leq GL(16,\mathbb{C})$ where $H = \{A_1 ...
2
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1answer
30 views

Similar matrices NOT over the complex numbers [duplicate]

We say that two matrices $A,B$ with complex entries are similar if and only if there exists an invertible complex matrix $P$ so that $A = P^{-1} B P$. Does $P$ always have to be a complex matrix? ...
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1answer
27 views

Find $ \text{rank}(T) $ and $ \text{nullity}(T) $.

If $ T: P_{2} \to P_{1} $ is defined by $$ T(p(x)) \stackrel{\text{df}}{=} p'(x) + p''(x), $$ find $ \text{rank}(T) $ and $ \text{nullity}(T) $.
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0answers
19 views

Is the action of the matrix UU^(t) always a projection? What can I say about I - UU^t?

The way I understand it is that: if U is orthogonal, i.e., its columns (or rows) form an orthonormal basis for the n-dimensional Euclidean (coordinate) space, then the matrix $UU^t$ is an orthogonal ...
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1answer
37 views

Product of upper-triangular matrices.

** I´m trying to solve this problem, but I don´t now how to start, I think could be by induction but I´m not sure. ** Let $n$ be a positive integer and let $F$ be a field. Let $A_1, . . . , ...
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0answers
17 views

Prove $U_1⊕…⊕U_m$ is finite-dimensional and $dim U_1⊕…⊕U_m = dimU_1+…+dimU_m$

Suppose $U_1,...,U_m$ are finite-dimensional subspaces of V such that $U_1+...+U_m$ is a direct sum. How to apply $dim(U+V)=dimU+dimV-dim(U∩V)$ to more than 2 subspaces? Please help me with a rigorous ...
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1answer
15 views

Does $AS=SB\iff f_A(\lambda)=f_B(\lambda)$?

Showing the converse is straightforward: $$B=S^{-1}AS\Rightarrow f_B(\lambda)=\det(B-\lambda I_n)=\det(S^{-1}AS-\lambda I_n)=\det(S^{-1}(A-\lambda I_n)S)\\=(\det S)^{-1}\det (A-\lambda I_n)\det ...
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0answers
9 views

A proof for a theorem related to rank and matrix product. [duplicate]

For all matrix $\mathbf{M} \in \mathbb{R}^{m,n}$ and $\mathbf{N} \in \mathbb{R}^{n,p}$, the inequality $\operatorname{rank}\mathbf{M} + \operatorname{rank}\mathbf{N} - n \leq ...
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3answers
32 views

Is any linear transformation with $\text{ker }(T)=\left\{\vec{0}\right\}$ an isomorphism?

I'm thinking no; for instance, $\exists \left\{\left.T:V\rightarrow W\right| \text{Im }(T)\neq W\right\}$. This seems counterintuitive, though. If such a $T$ with maximal rank exists, What would ...
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1answer
27 views

Sum of two vector subspaces [on hold]

V and W are vector subspaces $$ V = \left\{(x, y, z) \in \mathbb{R}^3, x + 2y -z = 0\right\} $$ $$ W = \left\{(x, x, x), x \in \mathbb{R}\right\} $$ Calculate V + W
2
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1answer
29 views

If $v_1,…,v_m$ are linearly independent, then the span $v_1+w,…,v_m+w$ has dimension $\ge m-1$

Suppose $v_1,...,v_m$ is linearly independent in $V$ and $w\in V$. Prove that $$ \dim (\operatorname{span}(v_1+w,...,v_m+w)) \ge m-1$$ It's an exercise in the book Linear Algebra Done Right. ...
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2answers
16 views

What is the maximum value of $\text{dim ker }A$, where $A$ is $n\times m$?

True or false: "If $A$ is an $n\times m$ matrix, then $\text{dim ker }A\leq n$" My gut intuitively tells me "no"$\,\Rightarrow$ if $m>n$, $\text{dim ker }A\leq m$. I can't think of a simple, ...
2
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1answer
13 views

What is the distinctive characteristic/structure of Polish Space?

I am trying to understand the geometric structure of Polish space. While reading I came up with the wikipedia link: http://en.wikipedia.org/wiki/Descriptive_set_theory and on the second paragraph of ...
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0answers
37 views

Why can't we eliminate $t$?

I'm reading Lang's Introduction to Linear Algebra. Here he says that it's not possible to eliminate $t$ in more dimensions. My problem is: Thinking about the method he gave in the first page, it seems ...
0
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1answer
12 views

Linear Independence of Vectors that are a Linear Combination of other Linearly Independent Vectors

Suppose v1,v2,v3 are linearly independent vectors in a vector space V and let w1 = v1 + av2 , w2 = v2 + av3, w3 = v3 + av1 for some a ∈ R. For what values of a are the vectors w1, w2 ...
4
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2answers
22 views

Does $\chi(g^{-1})=\overline{\chi (g)}$ hold for infinite groups

Let $\chi$ be the character of some representation $\rho:G \to GL(M)$ over $\mathbb C$. Suppose $G$ is a group, then $\forall g \in G$ of finite order $n$, $ \chi(g^{-1})=\overline{\chi (g)}$ ...
0
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1answer
20 views

Inner product space and orthogonality proof.

Why does this automatically mean that the sets are orthogonal? I am a little confused about this? How would I necessarily prove also?
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2answers
17 views

Showing $u_1, u_2, u_3$ is basis

Let $\{v_1, v_2, v_3\}$ be a basis for a vector space $V$. I want to show that $\{u1, u2, u3\}$ is also a basis where $u1 = v1, u2 = v1 + v2$ and $u3 = v1 + v2 + v3$ I wanted to use the standard ...
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2answers
32 views

Is every diagonal matrix the product of 3 matrices, $P^{-1}AP$, and why?

In trying to figure out which matrices are diagonalizable, why does my textbook pursue the topic of similar matrices? It says that "an $n \times n$ matrix A is diagonalizable when $A$ is similar to a ...
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3answers
16 views

What does the notation $[T]_{B^\prime \to B}$ mean?

Let $T:P_2 \to P_1$ be defined by $T(p(x))=p'(x) + p''(x)$ and let $B = \{1,x,x^2\} \text{ and } B'=\{1,x\}$. Find $[T]_{B\prime \to B}$ I do not understand the notation used when saying ...
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2answers
31 views

Orthonormal basis proof.

Let $\beta=(v_1,\ldots,v_n)$ be an orthonormal basis for $V$. Show that for any $x,y\in V$, $$\langle x,y\rangle=\sum_{i=1}^n \langle x,v_i\rangle \overline{\langle y,v_i\rangle}$$ How ...
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0answers
37 views

Hierarchy of mathematical jargon of algebra.

I am just beginning to learn algebra and having difficulty in understanding all the words, space, topology, linear space, Polish space, normed space etc etc. So I was wondering if we can have a ...
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1answer
10 views

Questions about orthogonal subspace proof.

I'm having a hard time grasping this intuitively, much less showing how to prove it. Any help would be appreciated. I don't get why a vector orthogonal to a subspace would be in the space itself. ...
2
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2answers
20 views

Can we find a basis such that $[T]_{B^\prime}$ is a diagonal matrix?

Let $T:P_2 \to P_2$ be defined by $T(p(x)) = x\frac{dp}{dx} + \frac{dp}{dx}$ and $B = \{ 1,x,x^2 \}$. We can find that $[T]_B = \begin{bmatrix}0 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 ...
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0answers
15 views

Linear Transformation Similarity - Answer verifications

Let $B = \{1,x,x^2\}$ and $B^\prime = \{1, 1+ x, 1+ x + x^2\}$ and $T(p(x)) = p(x) - x\frac{dp}{dx}$. Find $[T]_B$ Find the transition matrix from $B^\prime$ to $B$. Find the transition ...
1
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1answer
43 views

Assume that there exists an $\alpha\in\text{Aut}(V )$ satisfying $\alpha^{−1} = \alpha^2 + \alpha$. Show that $\dim(V )$ is divisible by 3

I need some help with this problem: Let $V$ be a vector space having finite dimension over $\mathbb{Q}$ and assume that there exists an $\alpha\in \text{Aut}(V)$ satisfying $\alpha^{−1} = ...
1
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1answer
18 views

Let F be a field and let $A,B ∈M_{n×n}(F)$ be a commuting pair of matrices, where B is nonsingular. Is $(A,B^{−1})$ necessarily a commuting pair?

I´m trying to solve this problem, but I can´t, I don´t know how to start. Let F be a field and let $A,B ∈M_{n×n}(F)$ be a commuting pair of matrices, where B is nonsingular. Is $(A,B^{−1})$ ...
0
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1answer
18 views

let $α, β, γ, δ$ be endomorphisms such that $α − β$ and $α + β$ are automorphisms. Show that exist $ϕ$, $ψ$ such that $ϕα + ψβ = γ$, $ψα + ϕβ = δ$.

I need some help with this problem: Let $F$ be a field of characteristic other than 2. Let $V$ be a vector space over $F$ and let $α, β, γ, δ$ be endomorphisms of $V$ satisfying the condition that $α ...
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1answer
47 views

Determinant Calculation Issue

Solved..found my mistakes.Thanks David for pointing out the first one to made me realize the other problem in C. I was asked to calculate the determinant for the following matrix: \begin{matrix} ...
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1answer
23 views

Non-negative Eigenvalues

Show that if $A$ is an $n × n$ matrix and $A = B^tB$ then every eigenvalue of $A$ is non-negative. I'm confused how the $A = B^tB$ relates to eigenvalues.
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1answer
11 views

The inverse of a state space matrix

First of all, I would like to link this question to another one about the inverse of a state space representation: Inverse of State-space representation I understand the prove as given on the ...
0
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1answer
19 views

Finding Eigenvectors and Eigenvalues

Let there be a matrix $$\begin{pmatrix} 1 & 3 & 5 \\ 1 & 3 & 3 \\ 1 & 5 & 1 \end{pmatrix}$$ Give an example for a vector how is not an Eigenvector (and not zero) The ...
0
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2answers
20 views

Prove that $A$ and $A^t$ have the same eigenvalues. [duplicate]

Let $A$ be a square matrix. Prove that $A$ and $A^t$ have the same eigenvalues. The solution my lecturer uses is: Consider the characteristic polynomial \begin{align} P_{A^t}(x) &= ...
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1answer
20 views

Complete eigenvalues

I need to confirm if the eigenvalue given is a complete eigenvalue and also need to determine the dimension of the associated eigenspace. I know that an eigenvalue is complete if the geometric and ...
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1answer
34 views

Show that a nonzero $2 \times 2$ matrix $A$ such that $A^2 = 0$ is similar to $\begin{pmatrix}0&1\\0&0\end{pmatrix}$

Let $A$ be a $2 \times 2$ non-zero matrix such that $${A}^{2}=0.$$ How do I find an invertible matrix P such that $${P}^{-1}AP=\begin{bmatrix}0&1\\0&0\end{bmatrix} ?$$ Anyone? Please provide ...
2
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1answer
57 views

The only eigenvalue of $A \in {M_n}$ is $\lambda = 1$. Why is $A$ similar to $A^k$?

Suppose that the only eigenvalue of $A \in {M_n}$ is $\lambda = 1$. Why is $A$ similar to $A^k$ for each $k=1,2,3,\dots$?
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3answers
30 views

Find the number of distinct roots for polynomial of degree 3

Can anyone guide me on how to find the number of distinct roots for a given polynomial of degree $3$, what is the best approach? I have this example $$ f(x)=-{x}^{3}+3x-y $$ where y is a real number, ...
0
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1answer
22 views

Calculating the adjoint

I am having some trouble understanding the idea of cofactors and adjoints of matrices. From my understanding the adjoint of a matrix is the transpose of the matrix of cofactors? $A=\begin{bmatrix} 1 ...
1
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0answers
44 views

Let $\text{Rank}{(A - \lambda I)^k} = \text{Rank}{(B - \lambda I)^k}$. Why are $A$ and $B$ similar?

Let $A$ and $B \in M_n$ be two matrices such that $$\forall k=1,2,\dots,n,\ \forall \lambda\ \text{eigenvalue of $A$},\ \text{Rank}{(A - \lambda I)^k} = \text{Rank}{(B - \lambda I)^k}.$$ Why are $A$ ...
-1
votes
2answers
11 views

$A$ is similar to $cA$ for some complex scaler with $\left| c \right| \ne 1$.why dose all eigenvalue of matrix $A$ are zero?

Let$A \in {M_n}$ and $A$ is similar to $cA$ for some complex scaler with $\left| c \right| \ne 1$.why dose all eigenvalue of matrix $A$ are zero?Is this true that matrix$A$ is nilpotent?