# 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.

82,523 questions
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### How to Solve a AU=B System when Determinant of A=0

Let $$A=\begin{pmatrix} 1 & 5 & -1\\ -2 & -10 & 5\\ -2 & -10 & -1\end{pmatrix}$$ and $$B=\begin{pmatrix} 0 \\ 15 \\ -15\end{pmatrix}.$$ To find a vector $U$ such that $A$ ...
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### Stability of linear autonomous systems

Let $x'=Ax, A\in \mathbb{R^{n,n}}$, be a linear autonomous system. Denote by $\{ \lambda_j\}$ the set of the eigenvalues of $A$. I want to study its stability. There is this following fact that im ...
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### Big-o notation confused

In error analysis we say the error is of order $\epsilon$ if the error is less than or equal constant multiple of $\epsilon$. The question is: what is the benefit we have if we can multiply by any ...
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### Convergence of unit vector in dir. of $M^{k}*v$ to the principal eigenvector of M when $k\to \infty$ and M is symmetric

It is a standard fact that a square matrix $M$ of dimension $n$ has at most $n$ distinct eigenvalues, each of them a real number, and the sum of their multiplicities is exactly $n$. We will denote ...
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### Proving Euclidian Norm squared is equivalent to transpose times matrix for x in R^n

Apologies if this has been answered already but I can't seem to find an answer that I think answers my question (or at least one I understand). Anyways the question is, ...
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### Invariant Subspaces and Orthogonality

I'm struggling with a question on invariant subspaces. If someone could possibly help me out here that would be great. The question is as follows: Let $A ∈ M$ where $M$ is the set of $n \times n$ ...
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### When diagonalizing a matrix does the order of the eigenvectors matter?

Say I find the eigenvalues of some matrix $A$ and then use kernel computation to find an eigenbasis for each eigenvalue. Then $B = S^{-1}AS$, where $S$ is composed of the eigenbasis of each of the ...
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### What is the Geometric meaning of vector norm in Rn n>3

My question is related to the length of the vector , Sorry it may seem stupid for you as i come from engineering background not mathematics background For Vectors up to 3 dimensions (can be ...
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### T/F: If a subspace S has a basis B with 2 elements then S is a plane.

I believe the answer is yes, because even if the basis is in $R$2, then the basis would represent the xy-plane. For $R$3 and beyond, two elements of a basis should always form a plane Is my ...
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### Generalize “matrix multiplication” with an function/operator?

Newbie question. No idea if this make sense or where to ask. Matrix multiplication is defined as, “If A is an n×m matrix and B is an m×p matrix, their matrix product AB is an n×p matrix, in which the ...
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### What rotations are performed to produce this output on a Tesseract?

I'm writing a program that projects a tesseract in 3D on a 2D environment and I want to reproduce the rotation of this gif but I'm having difficulty grasping what rotations before projection I need to ...
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### Find An Eigenvalue & Eigenvector from a given Eigenvalue and eigenvector

I was given a 3x3 matrix and I was given a complex Eigenvalue of -2+6i. They also gave me the eigenvector [-6-2i] [1] [9i]. I have to find another eigenvector and eigenvalue that isn't given. I ...
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### Ellipsoid and linear transformation

I get confused in my algebra about this simple problem. The equation of a 3D centred ellipsoid can be written on a compact form as $$x^{T}A x=1$$ with $x\in \mathbb{R}^{3}$ et $A$ a matrix that ...
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### Polynomial of Linear Operator

Let $L$ denote a linear operator and $v\in V$. Does the expression $$c_0L^nv + c_1L^{n-1}v + \cdots + c_{n-1}L^1v + c_nL^0v = 0$$ has a special name and what properties are known? For example, I know ...
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### Geometry of Solution Set and Matrix

Hello all. If wrote this system as a matrix, I would recognize it as a 3x5 matrix, but what I am not confident on is the fact that there are four variables, with constants on the other side. My gut ...
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### Confused about finding non-brute force way to solve for matrix to the 2019th power

I am attempting to solve this problem, it has four parts. I solved part a (a trivial matrix problem), but the next three parts appear to be a bit confusing to me. I just would like some help getting ...
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### What do you mean by scaling a vector in a particular direction?

Find a 2 × 2 matrix $A$ such that the linear transformation $T : R^2 → R^2$ defined by $T(~u) =A~u$ geometrically scales by a factor of 2 in the direction of the vector $\left(1,2\right)^T$ and scales ...
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### What is the total number of distinct $m\times n$ matrices in row canonical form using only $0$s and $1$s?

Suppose that $A$ is an $m \times n$ matrix over a field $F$. What is the total number $N$ of the distinct matrices in row-reduced echelon form that are row equivalent to $A$ and that only have entires ...
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### Showing $\Omega = PP'$ implies $P^{-1}\Omega(P')^{-1} = I$, where $\Omega$ [on hold]

Similar as the title, how to prove that $\Omega = PP'$ implies $P^{-1}\Omega(P')^{-1} = I$, where $\Omega$ is a symmetric and positive definite matrix.
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### Find two vectors X, Y [on hold]

Let $$C=\begin{pmatrix} 1 & 2 & 3 \\ -1 & 1 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix}.$$ Find two vectors $X, Y \in R^3$ such that $X^TCY \neq Y^TCX$.
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### When do two vector spaces with same dimension have the same basis?

I was thinking of this case: two vector spaces - $\mathbb{R}^2$ over $\mathbb{R}$ and $\mathbb{C}^2$ over $\mathbb{C}$. Every basis in the first vector space is also a basis for the second vector ...
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### Gram-Schmidt orthogonalization process with specific dot product

I have three vectors $$v_1=(1,1,1)^T$$ $$v_2=(1,1,0)^T$$ $$v_3=(1,0,0)^T$$ and special dot product definition $$(\overline{(x_1,x_2,x_3)},\overline{(y_1,y_2,y_3)})=2x_1y_2+x_1y_1+2x_2y_1+x_3y_3$$ I ...
### Find $A(v_1+v_2)$ and $A(3v_1)$ given eigenvectors and eigenvalues
If $v_1=[-1;5]$ and $v_2=[-3;5]$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\lambda_1=-1$ and $\lambda_2=1$, find $A(v_1+v_2)$ and $A(3v_1).$ I managed to find $A,$ ...