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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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How do we find non-self-adjoint A and unitary U such that exp(iA) = U?

The following is a theorem: If $A$ is a self-adjoint matrix (i.e. $A^\dagger = A$), then $U = e^{iA}$ is a unitary matrix. This is easy to prove: $(e^{iA})^\dagger = e^{-iA^\dagger} = e^{-iA}$, with the last step a consequence of self-adjointness…
senshin
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What is an intuitive definition of the zero vector?

I'm learning now about vector spaces and subspaces, and one of three rules that determine if something is a subspace of a larger vector space is that it must contain the zero vector... but intuitively, I can't figure out why the zero vector wouldn't…
Mirrana
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Why does adding a row to another in an augmented matrix preserves the solution set?

I get why multiplying one equation by a constant, and swapping equations, preserves the solution set(s) in a system of equations. But what I can't wrap my head around is why adding two rows in an augmented matrix preserves the solution set. Is…
Gary Greenhorn
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Pseudo Inverse Solution for Linear Equation System Using the SVD

I read about the SVD theory and its usage for solving Linear Equation System. I saw many papers mentioning property of the solution yet no proof of it. The Property: The solution given the Pseudo Inverse ( $ V {\Sigma}^{-1} {U}^{H} $ ) minimizes…
Royi
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calculate the equations for lines tangent to a sphere

Just started self-studying linear algebra, and as such I have no teacher I can ask. Working my way through first-year university material for linear algebra from a dutch university, so I might use the wrong english terms here. If so, please tell me…
ChrisD
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linear Algebra Kernel and Image Proof

let there be S,T linear operators working in vector space $U$; $$ T,S : U \rightarrow U $$ Prove that:$$ Ker(ST)=U \Longleftrightarrow ImT \subseteq Ker(S) $$ Attempt at a Solution: Left to right Direction: $ Ker(ST)=U \Rightarrow \forall u\in U…
Bak1139
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Suppose that $U$ is a subspace of $V$. What is $U+U$?

Suppose that $U$ is a subspace of $V$. What is $U+U$? Why does $U+U = U?$ I want to think of this geometrically, say in $\mathbb{R^{3}}$ we have some random plane in space that intersects the origin. How is this subspace, when added to itself,…
St Vincent
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Relationship between the column space of a matrix $A$ and its non-free (pivot) columns

Given an $m\times n$ matrix $A$ with $m\leq n$, with the rank of $A$ being less than $n$, is it necessarily true that the columns in $A$ representing the free variables are linear combinations of the pivot columns? If I am to figure out the column…
Mirrana
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Task interpretation for Hoffman Kunze Linear Algebra exercise 1 (b) sec. 3.6

I don't understand the task from (b). Is it equivalent to : for every linear functional $f(x_{1}, ..., x_{n})=c_{1}x_{1}+...+c_{n}x_{n}$ on $F^{n}$ which satisfy $c_{1}+...+c_{n}=0$ there exists exactly one and unique functional that belongs to the…
shooting-squirrel
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Theorem 20 Hoffman Kunze Linear Algebra book Section 3.6

What I don't understand here is why is $h(\alpha)=0$ for all $\alpha$ in $N_{k}$. Is there a typo? In case there is not, could someone please detail that last step please?
shooting-squirrel
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Expressing T in terms of its adjoint for T normal

The question is to show that if $T$ is normal, there exists a unitary operator $U$ such that $T^{*}=UT$. My guess is that we use the polar decomposition of $T$- into a product of a unitary and positive operator- in some way, but am not sure how to…
Arundhathi
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Proof of rank nullity theorem

I read about rank nullity theorem (with proof) but then tried to prove it in different way. Please can you read my proof and tell me if it is correct? The rank nullity theorem: If $T:V\to W$ is a linear map between finite dimensional vector spaces…
blue
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characteristic roots of a matrix

I have a question from linear algebra which I need to understand over the vacation. If $A$ is a square matrix over $\mathbb C$ and $A^n=I$, then why is every characteristic root of $A$ an $n$-th root of $1$ in $\mathbb C$? I've learnt what a…
Amy
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Finding dimension of space

Let $V = M_n(\mathbb{C})$ and define $$ f(A,B) = n\cdot\mathrm{tr}(AB) - \mathrm{tr}(A)\mathrm{tr}(B)\ . $$ Find $\mathrm{dim} (S)$ where $$ S = \left\{ v \in V\ | \ \text{for all}\ u \in V, \ f(v,u) = 0\right\}\ . $$ I first tried to check some…
user91015
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Proof that row rank=column rank using orthogonality

I wanted to find a proof that row rank = column rank using orthogonality, and I came across the following proof: Let $A \in \Bbb R^{m \times n}$ be a matrix with row rank $r(A)=r$. Let $\{ x_1,x_2,...,x_r \}$ be a basis of the row space. We shall…
TreasureGhost
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