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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.

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Linear span of the empty set

Why is the span of the empty set defined to be $\{0\}$? It is known that the span of any nonempty set of vectors in a vector space $V$, gives a subspace of $V$, and it is stated in “Linear Algebra Done Right” by Axler, that to be consistent with…
Train Heartnet
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What is the relationship between the null space and the column space?

Just looking at some tutorial videos, I'm noticing somewhat of a trend... but it wasn't spelled out explicitly, so I'd like to verify if this theory of mine is correct... (forgive my horribly un-exact notation...) Given $m\in\mathbb{N}$ vectors in…
Mirrana
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Is the proof of this lemma really necessary?

To prove the Cayley-Hamilton theorem in linear algebra, my professor said that a lemma was necessary: Lemma: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{K}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b(t)[A-tI]$,…
Andy
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Linear dependency of polynomials question

I have to determine whether the polynomials $p_1(x)=2-x^2$, $p_2(x)=3x$, $p_3(x)= x^2 +x-2$ are linearly dependent or independent but I am not sure how to start. Anyone care to enlighten me? Also I have to find out if it spans $P^{(2)}$.
user66167
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Vector spaces. When in the real world are we checking if it's a vector space or not?

I am reading this text: When in the real world are we checking to see if sets are vector spaces or not? The examples above seem like really specific sets... Are there any places where we redefined scalar multiplication like this?
Jwan622
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Why can a system of linear equations be represented as a linear combination of vectors?

I was watching Gilbert Strang's first Linear Algebra lecture, and the very first thing he does is relating the standard view of a system of linear equations as lines -in $\mathbb{R}^2$ of course- (what he calls the row picture) with the notion of…
Matt24
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If $A^2$ and $B^2$ are similar matrices, do $A$ and $B$ have to be similar?

I know that the converse is true; that is, if A and B are similar matrices, then $A^2$ and $B^2$ are similar . However, I'm not sure about the reverse.
user230049
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Why is anti-symmetry a desirable quality in determinants?

I hear the determinant of matrix can be defined using 3 facts. 1. It is multilinear. 2. It is anti-symmetric. 3. It is scaled so the determinant of the identity is 1. But, I don't understand why anti-symmetric is there? Why do people want…
Rioghasarig
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Eigenvalues in terms of trace and determinant for matrices larger than 2 X 2

The eigenvalues of a $2\times2$ matrix can be expressed in terms of the trace and determinant. $\lambda_\pm = \frac{1}{2}\left(\textrm{tr} \pm \sqrt{\textrm{tr}^2-4\det}\right)$ Is there a similar formula for higher dimensional…
mac389
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$\ker T\subset \ker S\Rightarrow S=rT$ when $S$ and $T$ are linear functionals

I would like only a hint to the following exercise: Let $V$ be a vector space over the field $K$, and $T$, $S$ linear functionals on V such that $Tv=0\Rightarrow Sv=0$. Prove that there exists $r\in K$ such that $S=rT$. I know how to prove this when…
Weltschmerz
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Wrong but fun and/or useful "proofs" in linear algebra

I wonder if anyone can share wrong but useful and/or fun proofs in linear algebra. It can not only be fun, but also useful for someone who learns the subject. You are welcome to add explanations but please hide them in order not to spoil the…
Aleksandr Kalinin
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Find the rotation axis and angle of a matrix

$$A=\frac{1}{9} \begin{pmatrix} -7 & 4 & 4\\ 4 & -1 & 8\\ 4 & 8 & -1 \end{pmatrix}$$ How do I prove that A is a rotation ? How do I find the rotation axis and the rotation angle ?
Kasper
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When does $V = \ker(f) \oplus \operatorname{im}(f)$?

For a vector space $V$ and a linear operator $f : V \to V$, under what conditions does $V = \ker(f) \oplus \operatorname{im}(f)$? Is it always true, or only in special cases? Edit: $V$ is finite dimensional.
Red
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What does it mean to work without a basis?

When reading proofs or definitions on Wikipedia, I'm accustomed to seeing both a basis-dependent discussion and basis-free discussion. Take, for example, this page on the tensor product, which has a "free vector space" discussion alongside a matrix…
Dragonsheep
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A finite-dimensional vector space cannot be covered by finitely many proper subspaces?

Let $V$ be a finite-dimensional vector space, $V_i$ is a proper subspace of $V$ for every $1\leq i\leq m$ for some integer $m$. In my linear algebra text, I've seen a result that $V$ can never be covered by $\{V_i\}$, but I don't know how to prove…
rhenskyyy
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