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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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Cayley-Hamilton...

Say $A$ is a square matrix over an algebraically closed field. Say $m$ is the minimal polynomial and $p$ is the characteristic polynomial. Of course C-H implies that $m|p$. Conversely, if we can show $m|p$ then C-H follows; the question is whether…
David C. Ullrich
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A normal, idempotent linear operator must be self-adjoint

I have been trying to solve this problem for quite a while. I am still unsure of whether any of the avenues I have pursued have been of any use. Any advice will be much appreciated. Question: Let $V$ be a finite-dimensional inner product space, and…
providence
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Is $\mathbb{R}$ a vector space over $\mathbb{C}$?

Here is a problem so beautiful that I had to share it. I found it in Paul Halmos's autobiography. Everyone knows that $\mathbb{C}$ is a vector space over $\mathbb{R}$, but what about the other way around? Problem: Prove or disprove: $\mathbb{R}$ can…
Potato
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Distance between two hyperplanes

I have two parallel hyper planes $$a^Tx=b_1,a^Tx=b_2$$ where $a \in \mathbb{R}^n, x \in \mathbb{R}^n ,b \in \mathbb{R}$ and I want to find the distance between the two. I have read that the distance between the two hyperplanes is also the distance…
Wanderer
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Show that the set of all symmetric, real matrices is a subspace, determine the dimension

Question: Let $V \subset M(n,n,\mathbb{R})$ be the set of all symmetric, real $(n \times n)$ matrices, that is $a_{ij} = a_{ji}$ for all $i,j$. Show that $V$ is a subspace of $M(n,n,\mathbb{R})$ and calculate dim$(V)$. My attempt so far: First part:…
ghshtalt
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Prove the determinant is the product of its diagonal entries

Prove that the determinant of an upper triangular matrix is the product of its diagonal entries. What I have so far: We will prove this by induction for an $n \times n$ matrix. For the case of a $2 \times 2$ matrix, let $A= \left(…
EmaLee
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Do the algebraic and geometric multiplicities determine the minimal polynomial?

Let $T$ denote some linear transformation of a finite-dimensional space $V$ (say, over $\mathbb{C}$). Suppose we know the eigenvalues $\{\lambda_i\}_i$ and their associated algebraic multiplicities $\{d_i\}_i$ and geometric multiplicities…
Ch Zh
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Linear Algebra: Preserving the null space

What does it mean when a book says that row operations preserve the null space? And why should that be true? I have read that row operations are equivalent to multiplying a vector on the left by an invertible elementary matrix. And I think I…
mcrocker
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Are all symmetric matrices ​invertible?

Is any symmetric matrix ​​invertible? I'm trying to prove this theoretical question, but I don't know what I need to do. I apologize for the simple question, but I'm in doubt and need clarification.
Ronny Portillo
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Every endomorphims is a linear combination of how many idempotents in infinite dimensions?

Every endomorphism of a finite-dimensional vector space is a linear combination of at most three idempotents, and the constant three is best possible, as Clément de Seguins has shown in this paper. On the other hand, Georges Lowther has shown in a…
Ewan Delanoy
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Prove: Square Matrix Can Be Written As A Sum Of A Symmetric And Skew-Symmetric Matrices

Let $C^{n \times n}$ be a square matrix. Prove that $$C=\frac{1}{2}(C+C^T)+\frac{1}{2}(C-C^T)$$ What I have manage so far is: a. Let $S$ be a Symmetric Matrix so $S=C+C^T$ b. Let $N$ be a Skew-Symmetric Matrix so $N=C-C^T$…
gbox
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Proof that $\dim(U \times V) = \dim U + \dim V$.

The following theorem in Serge Lang's Linear Algebra is left as an exercise, namely, Let $U$ and $V$ be finite dimensional vector spaces over a field $K$, where $\dim U = n$ and $\dim V = m$. Then $\dim W = \dim U + \dim V$, where $W = U \times V$,…
user38268
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What exactly is standard basis?

I am confused about the difference between coordinates and basis. My confusion is following: Let $e_i$ denote the standard basis and $v_i$ denote a non-standard basis of a finite $n$-dimensional vector space $V$. Then $e_i = (\delta_{ij})$ (all…
blue
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Intersection of kernels and linear dependence of linear maps

Let $f_1,...,f_n,f: V \to W$ be linear maps of $K$-vector spaces. If $W=K$ it's known that $f$ is linear dependent from $f_1,...,f_n$ iff $\;\;\bigcap_{i=1}^n \ker(f_i) \subseteq \ker(f)$. Question: Is this statement true for general $W$ ?…
Ralph
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$A$ is an invertible $n\times n$ matrix, where $n$ is an even number. Given that $A^3+A=0$, calculate $\det(A^4)$. Is there too much information?

$A$ is an invertible matrix with $n$ columns and $n$ rows, where $n$ is an even number. We are given that $A^3+A=0$ and we need to calculate $\det(A^4)$. Here is my solution: $$A^3+A=0 \implies A^{-1}(A^3+A)=0 \implies A^2=-I \implies A^4=I…
Omer
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