# Tagged Questions

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### Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
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### vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
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### $T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ and $<A,B> = Tr(AB^t)$

Let $V = M_{n \times n}(R)$ with the inner product $<A,B> = Tr(AB^t)$, and $T$ the linear operator given by $T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ . How can i ...
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### Eigenvector shared by two endomorhisms

I am guessing if the following fact is true: Let be $V$ a finite vector space above a field $K$. Let $f, g$ be two endomorphisms of $V$ with $f g = g f$. We assume that both $f$ and $g$ have got at ...
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### linear combination vectors into one vector

write $x(a_1,a_2,a_3)+y(b_1,b_2,b_3)+z(c_1,c_2,c_3)$ as $Y(a_1,a_2,a_3,$ $b_1,b_2,b_3,$ $c_1,c_2,c_3)$ ^as a 3x3 matrix for a suitable Y?
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### Dimension of vector space (count 0 or not)

Do you count 0 in the dimension of a vector space? Eg. If $V_\lambda$ is the eigenspace of a certain function $f$, which has eigenvectors corresponding to $\lambda$ of $v_1, v_2, v_3$ then the basis ...
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### Find eigenvector of the linear operator

Task is to find an eigenvector of the following linear operator: $f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$. I know how to find eigenvectors ...
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### Vector Space-Linear Transformation (Multiple Choice)

I was trying to solve the following problem from a competitive exam paper. Let $A$ be a nonzero linear transformation on a real vector space $V$ of dimension $n$. Let the subspace $V_0 \subset V$ be ...
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As a part of a larger proof, my text claims that if $$A\begin{bmatrix}u_1&u_2\\ \end{bmatrix}=\begin{bmatrix}u_1&u_2\\ \end{bmatrix}\begin{bmatrix}\lambda&1\\0&\lambda\\ ... 1answer 49 views ### Existence of invariant subspaces of \mathbb C^n and \mathbb R^n Exercise 6.15 from Dym's Linear Algebra in Action: Let A \in \mathbb R^{n\times n} and suppose that n \geq 2. Show that: (1) There exists a one-dimensional subspace U of \mathbb ... 1answer 62 views ### Invariant subspaces for endomorphisms with associated Jordan matrices I would like to know which are the invariant subspaces for the endomorphisms f1, f2, f3, f4, f5 from vector space V that have the next associated Jordan matrices: J1 = \left( ... 2answers 32 views ### Proving eigenvalue \lambda is a root for an annihilating polynomial p(t) I have the next statement I need to prove: Let \lambda be an eigenvalue of a endomorphism f and p(t) an annihilating polynomial of f. Prove that \lambda is a root for p(t). Thank you ... 1answer 49 views ### Proving subspaces are invariant for different statements [closed] I would like to know how to prove the next statements regarding invariant subspaces: Statement 1: f and g are endomorphisms from a vector space V. If f and g commute, then subspaces ... 0answers 101 views ### A basis for the column space of a real matrix Let A be a real square matrix, and let its column space be$$\mathrm{col}(A)=\{y\in\mathbb{C}^n:y=Ax\text{ for some } x\in\mathbb{C}^n\}.$$Under what conditions is \mathrm{col}(A) spanned by ... 1answer 22 views ### sum of two matrices question given condition How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also? 1answer 42 views ### Invariant subspaces using matrix of linear operator I am attempting the following problem but stuck at some parts: How does one find the (2 dimensional) subspaces that are invariant under A for$$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 &2 ...
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Let $T\in L(V,V)$, and let $\{v_1,v_2,\dots,v_n\}$ be a basis of $V$ consisting of eigenvectors of $T$, belonging to eigenvalues $a_1,a_2,\dots,a_n$ respectively. Then $Tv_i=a_iv_i$. Prove that ...
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### Stochastic matrix problem

A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to 1. Let $A$ be any (general) 2x2 stochastic matrix. a) Show that one of the ...
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### Rank of a large matrix

Suppose i want to calculate rank of a large $N\times N$ matrix having only $0$ and $1$ which is represented by $M$ segments each segment here is depicting that for a segment [LEFT,RIGHT] the row of ...
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Let $v=\begin{pmatrix}v_1\\v_2\\v_3\\v_4\end{pmatrix}$ be a nonzero column vector in $\Bbb R^4$ and let $A=vv^T$. Find the eigenvalues of $A$ There must be a easier way rather than calculate it ...
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### How to find vector that is orthonormal on other.

I have to generate a Q matrix for Schur decomposition and I have the first column, let's it is the following: \begin{bmatrix}1/âˆš3\\1/âˆš3\\1/âˆš3 \end{bmatrix} Now I need to find the second column that ...
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### Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
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### All Invariant Subspaces of a Linear Transformation

I got this problem: Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that all it's eigenvalues are 1, 2 and 3 and the corresponding eigenvectors are $v_1, v_2$ and $v_3$ ...
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### Variations in math to implement three-dimensional space?

Backstory: So I was researching topics, and found that 3-D game programming often markets itself with linear algebra. As a philosopher of math I decided to dig further into this and determine if ...
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### Prove the operators $T+U$ and $U$ have the same eigenvalues where $T$ is nilpotent

Let $V$ be an $n$-dimensional vector space on $\mathbb{C}$, and $T$ a nilpotent operator on $V$. Let $U$ be in $L(V)$ s.t. $UT = TU$. Prove that the operators $T+U$ and $U$ have the same eigenvalues. ...
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### Eigenvectors are linearly independent?

Theorem: Eigenvectors corresponding to distinct eigenvalues are linearly independent. Could someone give me a geometric interpretation of the theorem? Thanks!
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### Intuition behind independence of eigenvectors?

Theorem 6.14: Eigenvectors corresponding to distinct eigenvalues of A are linearly independent. My prof already gave us a proof of the theorem, so I'm not looking for another one. Could someone ...
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### Eigenspace and polynomials?

My prof introduced us to eigenvectors and eigenvalues today. He then gave us the following theorem: Theorem 6.6: Let $A$ be a square matrix, let $\gamma$ be an eigenvalue of $A$ with multiplicity ...
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### Eigenvector and Its Span

Let $V$ be a vector space over the field $F$ and let $T$ be a linear transformation from $V$ to $V$. Let $v\in V$ such that $v\neq 0$, let $W=span\{v\}$. Prove that if $T(W) \subset W$, then $v$ is an ...
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### Can you factor out vectors?

My prof introduced eigenvalues to us today: Let $A$ be an $n \times n$ matrix. If there a scalar $\lambda$ and an $n-1$ non-zero column vector $u$, then $$Au = \lambda u$$ then $\lambda$ ...
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### How to show that the limit of sequence of eigenvectors (same eigenvalue) is also an eigenvector?

Let $H$ be a continuous Hermitian operator on an infinite dimensional Hilbert space. Also, let $f_n$ be a sequence approaching $f$ as $n\to\infty$, where each $f_n$ is an eigenvector of the same ...
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### Disjoint Gershgorin disks $\Rightarrow$ each contains exactly one eigenvalue

It is an exercise in Peter Lax's book Linear Algebra that if all the Gershgorin disks $$D_i := \{z\in \mathbb{C} : |a_{ii} - z| \leq \sum_{i \neq j} |a_{ij}|\}$$ are disjoint, then each disk must ...