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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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There exist an infinite subset $S\subseteq\mathbb{R}^3$ such that any three vectors in $S$ are linearly independent.

Could anyone just give me hint for this one? There exist an infinite subset $S\subseteq\mathbb{R}^3$ such that any three vectors in $S$ are linearly independent. True or false?
Myshkin
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Why does polynomial factorization generalize to matrices

I'm reading about linear algebra and I came across with the following theorem where I have a problem convincing myself: Theorem 2.1 $\,$ Every linear operator on a finite-dimensional complex vector space has an eigenvalue. Proof: To show that $T$…
jjepsuomi
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How does the dot product convert a matrix into a scalar?

I am learning linear algebra, and I am a bit confused by the dot product and how the answer to the process turns out to be a scalar rather than a matrix. For $2$ vectors with $2$ components, I learned that dot product is equivalent to a $1 \times 2$…
treeorriffic
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'Free Vector Space' and 'Vector Space'

In this very nice book, author has defined vector space as set of functions $f : S \rightarrow F$ where $S$ is a finite set and $F$ is a field. It turns out that this definition is closely resemble definition of Free Vector Space. There are one or…
Dilawar
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6 answers

Prove that $n+1$ vectors in $\mathbb{R}^n$ cannot be linearly independent

I was looking for a short snazzy proof on the following statement: n+1 vectors in $\mathbb{R}^n$ cannot be linearly independent A student of mine asked this today morning and I couldn't come up with a proof solely from the definition of linear…
Wyatt
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23
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7 answers

What is the use of the Dot Product of two vectors?

Suppose you have two vectors a and b that you want to take the dot product of, now this is done quite simply by taking each corresponding coordinate of each vector, multiplying them and then adding the result together. At the end of performing our…
Antonio
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5 answers

Why can I solve an impossible equation using linear algebra?

I am currently learning matlab and linear algebra side by side and I stumbled upon this example from mathworks A = [1 2 0; 0 4 3]; b = [8; 18]; x = A\b x = 3×1 0 4.0000 0.6667 which in my mind translates to $$ A = \left[ \begin{matrix} 1 & 2…
linker
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23
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Can vectors be inverted?

I wish to enquire if it is possible to solve the below for $C$. $$B^{-1}(x-\mu) = xc $$ Here obviously $B$ is an invertible matrix and both $c$ and $\mu$ are column vectors. Would the solution be $$x^{-1}B^{-1}(x-\mu) = c $$ is it possible to invert…
Comic Book Guy
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4 answers

Intuitive explanation of why $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$

I'm having a hard time truly understanding the meaning of $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$ where $V$ is the domain of a linear transformation $T:V\to W$. I've used this equation several times in many problems, and I've…
Daniel
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The eigenvalues of a block matrix of the form $\pmatrix{B&C\\0&D}$

It is known that all $B$, $C$ and $D$ are $3 \times 3$ matrices. And the eigenvalues of $B$ are $1, 2, 3$; $C$ are $4, 5, 6$; and $D$ are $7, 8, 9$. What are the eigenvalues of the $6 \times 6$ matrix $$\begin{pmatrix} B & C\\0 &…
Scorpio19891119
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Linear independency before and after Linear Transformation

If we are given some linearly dependent vectors, would the T of those vectors necessarily be dependent (given a transformation from $R^n$ to $R^p$)? And if we are given some linearly independent vectors, would T of those vectors necessarily be…
LinAlgStudent
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If every eigenvalue of $A$ is zero, does this mean $A$ is a zero matrix?

If every eigenvalue of $A$ is zero, show that $A$ is nilpotent. I got this question as my homework. I am just wondering if every eigenvalue of $A$ is zero, then $A$ is zero, why bother to prove $A$ is nilpotent.
Shannon
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Vector spaces of the same finite dimension are isomorphic

Suppose that V and W are vector spaces with the same dimension. We wish to show that V is isomorphic to W, i.e. show that there exists a bijective linear function, mapping from V to W. I understand that it will suffice to find a linear function that…
Bysshed
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Method to reverse a Kronecker product

Let's say I have two simple vectors: $[0, 1]$ and $[1, 0]$. Their Kronecker product would be $[0, 0, 1, 0]$. Let's say I have only the Kronecker product. How can I find the two initial vectors back? If my two vectors are written as : $[a, b]$ and…
glmxndr
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$\sum_{k=0}^{100} a_k x^{k}=0, a_i \in \mathbb{Z}$ How many equations?

Let $x^{100}+a_{99} x^{99}+a_{98} x^{98}+ \dots +a_{1}x +a_{0}=0, \ \ a_{100}=1, a_i \in \mathbb{Z}$, is an algebraic equation with integer coefficients. Assume that all (100 with multiplicity) roots are positive real numbers: $0 < x_1\leq x_2\leq…
user59928
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