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Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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5 views

Determining the rank, nullspace and image of a matrix with one unknown

Consider the matrix: $$A=\begin{pmatrix}x &0&2&2 \\-2&2&-2&2\\2&2&-2&2\\2&0&2&-2 \end{pmatrix}$$ Determine $\text{rank(A)}$ as well as a basis for the ...
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0answers
16 views

(Solved) Question About Schur's Theorem

Schur's theorem is about Upper triangular representation of an linear operator. This is from Linear Algebra Done Wrong by Sergei Treil. (1) Schur's Theorem. Let $A: V \rightarrow V$ be an operator ...
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1answer
9 views

Conditions of a set of vectors, so that a specific linear map is injective/surjective

Let $m \in \Bbb N$ and $v_1,\dots,v_m \in V$ be distinct vectors. Furthermore let $A\colon =\{v_1,\dots,v_m\} $ and $$T: \Bbb K^m \to V \\ T(x_1,\dots,x_m) = x_1v_1+\dots+x_mv_m$$ be a linear map. ...
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1answer
19 views

Equivalence of a scalar to a vector:

I just got myself into this position while studying matrix operation, I am sure others must have been there before . If A is a matrix of dimension (m,n) and k is a scalar then ( A + k ) gives the ...
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3answers
30 views

How can I solve the linear recurrence problem $f(n)=f(n-1)+3 \cdot f(n-3)+2n$ using matrix exponentiation when $ f(1)$ , $f(2)$ and $f(3)$ are given.

The porblem is $f(n)=f(n-1)+3f(n-3)+2n$. I solved $f(n)=f(n-1)+3f(n-3)$ and adding summation of $2n$ upto $n$. But this is wrong. It requires too much of pre calculation. I tried this problem based on ...
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2answers
24 views

how do I check linear independence of m vectors, given the m-1 first vectors are independence?

I have a set of m-1 vectors of size n, they are independence. I want to add another vector and to check if the m vectors are still independence. It is known m < n. I am looking for a good algorithm ...
2
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1answer
21 views

A question on linear independence and dependence

I have been trying to solve the second half of the following problem but so far I have not succeeded so far: Let $\alpha_{1},\ldots,\alpha_{n}$ be $n$ elements in a vector space $V$ over $\mathbb{...
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0answers
12 views

Multiplication of cyclic diagonals.

We denote by $D_k$ the set of all matrices of the form $A=(a_{i,j})_{i,j=0}^{n-1}$ , such that $a_{i,j}\neq 0$ for $i-j=k$ or $k-n$, and $k=1,2,\cdots n-1$. Then we have to show that if $A\in D_k$ and ...
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0answers
21 views

eigenspace corresponding to eigenvalue 1?

Let $A$ be in $GL(2,n)$ means $n$ by $n$ invertible matrix with entries in $Z_2$, If $A$ has eigenvalue $1$, Is there something we can say about structure of matrix $A$ or about $\dim(\ker(I-A))$ or ...
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1answer
22 views

inverse matrix search algorithm

For ex, $$A^{-1}\cdot A=\left[\begin{matrix}x_1 & x_2\\x_3 & x_4\end{matrix}\right]\cdot\left[\begin{matrix}1 & 2\\1 & 3\end{matrix}\right]=\left[\begin{matrix}1 & 0\\0 & 1\...
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0answers
14 views

Showing $\{ \alpha _{i} \}$ forms a basis of $V$

In a vector space $V$ over $\mathbb{F}$, assume $\alpha _{1} , \ldots , \alpha _{n}$ are linearly independent but that $\beta , \alpha _{1} , \ldots , \alpha _{n}$ are linearly dependent for each $\...
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1answer
52 views

Prove all roots of $p_n(x)-x$ are real and distinct

Given a polynomial series $\{p_n(x)\}_{n=1}^{\infty}$ in $\mathbb{R}[X]$ with initial value $p_1(x)=x^2-2$. And $p_k(x)=p_1(p_{k-1}(x))=p_{k-1}(x)^2-2,\;k=2,3,\cdots$. Prove that for each integer ...
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1answer
47 views

Show that $\exp(C^{-1} AC ) = C^{-1} \exp(A C)$

Show that $\exp(C^{-1} AC) = C^{-1} \exp(A C)$ for any matrices $A \in L_{n}(\mathbb{R})$ and $C \in GL_{n}(\mathbb{R})$. The hint of the question is given below: Consider the linear operator $\...
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0answers
21 views

On $\bigg[\begin{matrix}E&O\\O&F\end{matrix}\bigg],B_b=\left[\begin{matrix}2b&-1&0&-1\\0&b&0&0\\0&-1&0&-1\\0&1&0&b\end{matrix}\right]\in M_4(\Bbb R)$

Consider the $4\times4$ matrices $A=\bigg[\begin{matrix} E&O_2\\O_2&F \end{matrix}\bigg]$, where $E,F$ are any nilpotent $2\times2$ matrices, and $B_b=\left[\begin{matrix}2b&-1&0&-...
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0answers
7 views

Vector spaces and homogenous linear equations system

For every vector space $R^n$ there is homogenous system of linear equations whose set of all solutions is isomorphic to $R^n$ This should obviously be true, but I am not sure I understand intuition ...
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1answer
29 views

Proof of all polynomials with degree n can for a basis

Prove that {$1, x, x^2, x^3, ... , x^n$} form a basis of $P_n$ ($n$ is a non-negative integer) , the space of the polynomials of degree $n$. Using a general proof.
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1answer
40 views

Why does $\dim W = \dim V$ only if $W = V$ for a vector space $V$ and its subspace $W$? Removing one element from W renders this false

Let $W$ be a subspace of a finite-dimensional vector space $V$. Then $\dim W = \dim V$ if and only if $W = V$. Why is this? I understand the proof I was given (which uses the fact that any set of ...
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2answers
25 views

How to find base of U+kerf+W?

Let be $U=<(1,0,0),(0,1,0)>, W=<(-3,1,1)>,Kerf=<(-3,1,1)>$ three subspaces in $\mathbb R^3$. If I have this $3$ generating systems, how can I get a base of $U+Kerf+W$?
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1answer
11 views

Is there any relation between SVDs of two matrices with same range?

Let $A$ and $B$ be two symmetric and positive semidefinite matrices with the same size. Further, assume that $A$ and $B$ share the same column space (i.e., $\mathcal R (A) = \mathcal R (B)$ ). Is ...
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0answers
12 views

Why don't we generalize tensor to several different spaces

Tensor of $(r,s)$ type is defined as the dual of $(V)^r\times (V^*)^s$, which is the tensor product $(V^*)^r\otimes(V)^s$. I am wondering why we don't generalize this definition to $r$ different ...
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1answer
30 views

Equality concerning the norm of rows of a resolvent matrix.

This problem showed up on UCLA's basic exam for Fall 2018: Let $X$ be an $n \times n$ symmetric (real) matrix and $z \in \mathbb{C}$ with $\text{Im } z > 0$. Define $G = (X - zI)^{-1}.$ Show ...
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2answers
18 views

What is the significance of the fact that a set in a vector space is linearly dependent if it contains finitely many linearly dependent vectors?

A set $S$ in a vector space $V$ is linearly dependent if it contains finitely many linearly dependent vectors. What is the significance of this definition? Obviously for finite sets, this is no ...
4
votes
4answers
73 views

Linear independence of $1, e^{it}, e^{2it}, \ldots, e^{nit}$

Definition: Let $C[a,b]$ be the set of continuous $\mathbb{C}$-valued functions on an interval $[a,b] \subseteq \mathbb{R}$ with $a < b$. Claim: In $C[-\pi, \pi]$, the vectors $1, e^{it}, e^{...
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1answer
12 views

conditions of an inequation

I need to find the solutions of this inequation. I put the conditions and then I found x for and I obtained which should be intersected with x from the conditions and I obtain in the final but it'...
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3answers
36 views

True or False: Entries on the main diagonal of matrix A

Q. If $A = [a_{ij}]$ is an $m \times n$ matrix which satisfies $A^T = -A$, then the entries on the main diagonal of $A$ are all equal to $0$. I don't see how $A^T = -A$ can be true for a $m \times n$ ...
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2answers
59 views

I didn't understand 3blue1brown's video on inverse matrices, where a matrix has no inverse if its determinant is zero

I was watching 3blue1brown's video on inverse matrices, and I didn't understand what he said about the case where a matrix $A$ has no inverse when $\det(A)=0$. Considering $Ax=v$ where $x$and $v$ are ...
3
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4answers
133 views

What is relationship between scalar and vector product?

I am intrested in relationship between scalar and vector product in $\mathbb{R}^3$; I am going to give definitions which I will use in my question. Scalar product - function $\cdot:\mathbb{R}^3 \...
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0answers
20 views

Theoretical questions about characteristic polynomials, diagonalizability, rank and similarity

I am new to Linear Algebra and would like some feedback regarding my answer to the following question A and B are two square matrices of order n. Prove or refute (with a counterexample) the following ...
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1answer
18 views

Diagonalizability of matrix over C and R respectively

I am new to linear algebra, and was asked the following question: $A$ is a matrix of order n x n, $n \geq 2$, with characteristic polynomial $p(λ)=λ^n-1$. Is $A$ diagonalizable over R? over C? If $A$ ...
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0answers
30 views

please tell solution to me [on hold]

Let A be an improper orthogonal matrix then adjoint A is equal to 1)A 2)transposeA 3)-A 4) -(transposeA)
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1answer
13 views

Eigenspace for a linear transformation + diagonalizability

I am new to linear algebra, and was asked the question below. I am just looking for some feedback regarding my proposed answer. $T:R_4[x] \to R_4[x]$ is a linear transformation defined by $T(p(x))=p(...
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0answers
79 views

Eigenvalues keep giving trivial solutions everytime.

I am trying to find the eigenvalues of this Eigen BVP. $\mu$ is the eigenvalue parameter $$ \lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \...
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0answers
8 views

finding a base and dimension of a 3 x 3 matrix that represents all skew-symmetric matrices above the complex field [duplicate]

I'm suppose to find that matrix, and I was only given the final answer of the dimension which is 3. But everything I tried so far is leading me to one of the rows being zeroed and therefore the ...
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1answer
19 views

In linear algebra, how do we define direct sum between two sets if we choose an expression like A + B = A + B? [on hold]

in linear algebra, we define the direct sum between two sets like this : A +* B = if and only if (A + B = C and A ∩ B = {0}). How do we define A +* B = A +* B ? I mean : is it always true or ...
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0answers
16 views

What is the computational cost of reduced row echelon form (RREF) of a rectangular matrix A?

Assume that $A$ is an $m\times n$ matrix in $\mathbb{R}$ in which $m\leq n$. Is it possible to find the computational cost of the reduced row echelon form (RREF) of $A$?
2
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1answer
19 views

Infinite norm of a vector

While reading the book Numerical Linear Algebra by Trefethen and Bau, I came across the following example. The authors indicate that $\|J\|_{\infty} = 2$, however if I recall the definition of $\|\...
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1answer
49 views

Can the sum of the first $p$ factorials ever be a perfect power for $\ p>3\ $?

Has $$\sum_{j=1}^p j!=q^r$$ , where q,p,r are positive integers, and r > 1 , a solution ? I solved partially, if r is even, then RHS is a perfect square, and there is no doubt in that. Therefore, the ...
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2answers
13 views

Eigenvalues and eigenvectors in combined Linear Transformations

I am new to linear algebra, and cannot work out the following question, despite the fact that I have been thinking about it for a long while. Let $V$ be a linear space of n dimensions over R, and let ...
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0answers
18 views

proving if player 1 has a pure optimal strategy, player 2 should as well

Problem: Prove that if in a matrix game 2x2 if the player 1 has a pure optimal strategy, so has player 2 Attempt: Given: We know that player 1 has pure optimal strategy, meaning: $$P(x, \overline{...
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1answer
13 views

Diagonalizability in relation to characteristic polynomial and row equivalence

I am new to linear algebra, and am unsure re the following question: True or False? Let A and B be matrices of n x n. If A and B are diagonalizable and they have the same characteristic polynomial, ...
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0answers
15 views

Similar matrices if only difference is diagonalizable/non-diagonalizable

I am new to linear algebra, and am just looking for some feedback regarding the following solution: True or false? 1.$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ and $$\begin{pmatrix}2&0\\0&...
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0answers
15 views

conditions for an equation with parameter [on hold]

There is the following equation: and it's roots. I need to find m such that . I know that I have to put the following conditions: \begin{matrix}\Delta >0\\ -1<-\frac{b}{2a}<1\\ a\cdot f(-...
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2answers
27 views

Dimensional Vector spoace with injective surjective and bijective

I know there are similar questions on the internet, but I'm not getting it smart. i hope u can help me, Let $B,C$ finite dimensional Vector space with $\dim(C) = \dim(B)$ and $R \in Hom(B,C)$. Then ...
2
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0answers
18 views

Quickest way to find the eigenvalues of a binary hermitian matrix?

I have an 11x11 matrix and need to find its charactaristic polynomial (or eigenvalues). I would like to do this analytically but was wondering if there's any tricks that can be used due to its ...
0
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1answer
22 views

Algebra parameter

$x,y,z,u$ are variables and $a,b$ parameters $(a+1)y+x−(a+1)z−au=1$ $(a−1)x+(2a+1)z+(a−2)u=1$ $(−a−2)z+(a+2)u=b−2$ Now, next steps are these: If $a\neq−1,a\neq1,a\neq−2$: System is indefinite If ...
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0answers
16 views

Let $A$ be an improper orthogonal matrix… [on hold]

Let $A$ be an improper orthogonal matrix then adjoint of $A$ is equal to $$A$$ $$A^T$$ $$-A$$ $$-A^T$$
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2answers
25 views

Prove that product of two matrices is $I$

The pair of variables $(x, y)$ are each functions of the pair of variables $(u, v)$ and vice versa. Consider the matrices: $$ A=\left(\begin{matrix} \frac{\partial{x}}{\partial{u}}& \frac{\...
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votes
1answer
24 views

Why do row equivalent matrices need to have same number of rows?

If any two rows of a $3×3$ matrix A turn out to be same and if we remove that row and it turns out that this new matrix is row equivalent to a $2×3$ matrix B then can we say matrix A is row ...
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1answer
18 views

direct sum of $T$-invariant subspaces

Let $T:V \rightarrow V $ be a linear transformation. For $v \in V$ I construct the $I(v)=\mathrm{span} \{v,Tv,...T^{k-1}v\}$. If $\mathrm{dim} (I(v)) < \mathrm{dim}V$, I can choose a $u$ which ...
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0answers
21 views

Q.5 (ii) 2016 portion [on hold]

(https://i.stack.imgur.com/kjYNA.png) Need solution of Q.5 (ii) 2016 portion.send me solution