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.
80,453 questions
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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.
...
0
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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{...
0
votes
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 ...
1
vote
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 ...
0
votes
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\...
1
vote
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 $\...
6
votes
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 ...
0
votes
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 $\...
0
votes
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&-...
0
votes
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 ...
-3
votes
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.
0
votes
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 ...
-2
votes
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$?
0
votes
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 ...
-1
votes
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 ...
5
votes
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 ...
1
vote
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^{...
0
votes
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'...
0
votes
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$ ...
1
vote
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
votes
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 \...
1
vote
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 ...
0
votes
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$ ...
-6
votes
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)
0
votes
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(...
2
votes
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' + \...
0
votes
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 ...
-2
votes
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 ...
0
votes
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
votes
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 $\|\...
1
vote
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 ...
0
votes
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 ...
0
votes
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:
$$P(x, \overline{...
0
votes
1answer
13 views
Diagonalizability in relation to characteristic polynomial and row equivalence
I am new to linear algebra, and am unsure re the following question:
True or False?
Let A and B be matrices of n x n.
If A and B are diagonalizable and they have the same characteristic polynomial, ...
1
vote
0answers
15 views
Similar matrices if only difference is diagonalizable/non-diagonalizable
I am new to linear algebra, and am just looking for some feedback regarding the following solution:
True or false?
1.$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ and $$\begin{pmatrix}2&0\\0&...
0
votes
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(-...
0
votes
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
votes
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
votes
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 ...
-2
votes
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
$$A$$
$$A^T$$
$$-A$$
$$-A^T$$
0
votes
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{\...
0
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 ...
1
vote
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 ...
-6
votes
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
