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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24 views

What's the word for a number which is used to scale down a value?

I'm a programmer and I'm creating an API in which there is a parameter the user can pass in which scales down a value. So for example: ...
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2answers
19 views

Let $A$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?

Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
0
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0answers
15 views

Let $V$ be the vector space over $R$ composed of all polynomials in **R[X]** having degree less than 3

can someone help me with this problem please. Let $V$ be the vector space over $R$ composed of all polynomials in R[X] having degree less than 3 and let $W$ be the vector space over $R$ composed of ...
0
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0answers
9 views

Projection out of orthogonal matrices

Let A,B be orthogonal matrices of order $n \geq 2 $. $\det A = 1, \det B = -1$. There exist $a \in [0,1]$ such that $aA + (1-a)B$ is projection. I know that the claim above is false. I ...
1
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1answer
30 views

Isn't it a subspace?

I have a problem with the concept of subspace. Determine that following sets are subspaces of $R^2$. (1) $W = \{(a,b+1)|a,b \in R\}$ (2) $V = \{(a+2b,b+1)|a,b \in R\}$ I know $W$ ...
0
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1answer
4 views

Variety of maximal isotropic subspaces

Suppose that $V$ is a complex vector space of even dimension $2n$. Let $Q:V \times V \rightarrow \mathbb{C}$ a bilinear, non degenerate, simmetric bilinear form over the field of complex number. Set ...
0
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1answer
14 views

Second Derivative Test and Hessian for $f(x,y) = x^2 + y^2$.

My task was to find the critical points of the function $f(x,y) = x^2+y^2$, to then compute the Hessian, and to use the second derivative test to determine whether the critical points are local maxima ...
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1answer
10 views

interpolating and difference table, an old mid exam?!

For calculating divided (fraction) difference table for interpolating the points $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, how many fraction was used? ...
3
votes
2answers
41 views

Equivalence of $\|x\|_1\|x\|_{\infty}$ and $\|x\|_2^2$

Let $x$ be any complex $n$-vector and let $\|\cdot\|_p$ denote the usual $p$-norm. It is easy to show that $\|x\|_2^2\leq\|x\|_1\|x\|_{\infty}$ (Hölder's inequality). What I am rather interested in is ...
0
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0answers
9 views

Matrix Properties Reference

A lot of proofs I come across (working on stability of numerical methods) apply some property of a particular type of matrix. This includes, for example, the fact that the $L_{2}$ norm of a normal ...
0
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1answer
21 views

Matrix Change of Basis

guys. I'm not entirely sure how I'm not getting the right answer for this question. I'll try to explain what I've tried so far. I need to computer MB1->B2 and MB2->B1 B1 = {(0,0,1),(1,0,0),(0,1,0)} ...
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3answers
31 views

Tell me whether this Unknown Operation Exists.

I need to know whether the below unknown operation, denoted by $\boxplus$ exists. If $v_1=a \boxplus X$ and $v_2=b \boxplus X$, where $X$ is an identical value in both $v_1$ and $v_2$: equation (1): ...
0
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0answers
16 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?
4
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1answer
47 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 $n \times n$ ...
0
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1answer
14 views

Equation for adjoint transformation and proof.

I am really lost on this one. Any help would be appreciated. I'm very confused.
11
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8answers
429 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 ...
0
votes
1answer
22 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 ...
4
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3answers
40 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 ...
0
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0answers
28 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.
1
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1answer
24 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 ...
0
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0answers
5 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
votes
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? ...
0
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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 ...
0
votes
1answer
38 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, . . . , ...
0
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0answers
18 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 ...
0
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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 ...
1
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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 ...
1
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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 ...
-3
votes
1answer
29 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
votes
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. ...
0
votes
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
votes
1answer
14 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 ...
0
votes
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
votes
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
votes
2answers
24 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
votes
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?
0
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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 ...
0
votes
2answers
34 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 ...
1
vote
3answers
21 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 ...
1
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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 ...
1
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0answers
41 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 ...
0
votes
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
votes
2answers
21 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 ...
1
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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
vote
1answer
45 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
vote
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 $(A,B^{−1})$ ...
1
vote
1answer
25 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 ...
-2
votes
1answer
48 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} ...