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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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Proving this is a linear subspace

I need to prove the following is a subspace: Let $V$ be a set of vectors over $F=\mathbb{R}$, $V=\operatorname{Functions}(\mathbb{R} ,\mathbb{R})$ and $W$ is a subgroup of $V$ such that $$W=\{f\in V| \, f(x)=f(-x)\}$$ I'm not sure about the "close…
yuvalz
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Linear algebra on Fibonacci number

Consider the sequence $\{a_n\}_{n\ge 0}$ given by the recurrence relation $$a_0=1,\ a_1=-1,\ a_{n+1}=3a_n+10a_{n-1}\ \ \text{for } n\ge2$$ And I am asked to work out the closed form expression for an in the same fashion as the proof for Fibonacci…
Chloe
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Eigenvectors with many rational entries

Suppose that $A$ is an invertible $3 \times 3$ matrix with integral entries and that $v$ is an eigenvector corresponding to an irrational eigenvalue. Of course by scaling $v$ we may arrange that it has at least one rational entry. But is it…
Tim Piyim
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Why do we need a zero vector space?

Reading through some lecture notes, I found out the first example of a vector space is the zero vector space, which contains only the zero vector. I understand that it meets all the axioms of a vector space, but why do we need it? Why bother…
AnalysisShmalysis
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Proving $\chi$ fixes two points in the unit sphere.

Let $\chi:\mathbb{R}^{3}\to\mathbb{R}^{3}$ be an orthogonal transformation such that $\det(\chi)=1$ and $\chi$ is not the identity linear transformation. Let $S \subset \mathbb{R}^{3}$, be the unit sphere. Then how do we prove that $\chi$ fixes…
lahari
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Maximal Ideals in $C((0 ,1))$

For a set $S\subset\mathbb R$ let $C(S)$ denote the continuous real-valued functions on $S$. Describe the maximal ideals in $C((0,1))$. For $C([0,1])$ we know that maximal ideals are points in $[0, 1]$ and these are the only maximal ideals, all…
Lawrence Mano
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Jordan canonical form over $\mathbb{R}$

The theorem about Jordan canonical form states that for any operator $f:V\to V$ where $V$ is a vector space over $\mathbb{C}$ there is a basis such that the matrix of this operator in this basis is a union of Jordan blocks. And in my lecture notes…
RFZ
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How can a vector represent velocity and a position as well?

I'm relatively new to linear algebra. I've got a question about a question. I'm not looking for the exact answer (I'll try to find it by myself). I stumbled upon the following question : "*At 12:00 pm, a spaceship is at position $$\begin{pmatrix} 3…
aschiff
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Methodical way to form a basis

If I have a basis $(3,1,0,0,0),(0,0,7,1,0),(0,0,0,0,1)$ for a subspace of $\mathbb{R}^{5}$ and want to extend this collection of vectors to a basis for $\mathbb{R}^{5}$ is there any methodical, or algorithmic way to do this besides logically picking…
user736276
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Can a transcendent matrix have an algebraic spectrum?

Let $K$ be an algebraically closed field (e.g $\mathbb{A}$) and $K'/K$ a transcendent field extension (e.g. $\mathbb{C}/\mathbb{A}$). Let $A\in K'^{n\times n}$ be a matrix over K', which has at least one entry from $K'\setminus K$. Is it still…
Isidor Konrad Maier
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Solution of Matrix Equation---Positive Definite Matrix

Let $A,C$ be $n$-dimensional symmetric matrix, $A$ is negative definite, while $C$ is positive definite. Assume that $AX+XA+2C=0$ has a unique solution $X=B$, prove then $B$ is real, symmetric, and positive definite. How to prove it? I have no idea.…
XLDD
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Involution $\Rightarrow$ Hermitian & Unitary

I've been learning about unitary diagonalization and this question has been on my mind for a while, unanswered by the text I'm using. Would it be true to say that if a square matrix represents an involution, i.e. $A^2=I$ (or $A=A^{-1}$), then it is…
Dahn
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How to (dis-)prove that this symmetric matrix is P.S.D.?

for two random $n$-vectors $(X_1,X_2)$ define $$f(X_1,X_2)=\text{med}((X_1+X_2)^2)-\text{med}((X_1-X_2)^2)$$ Now define the matrix $U$ with entries $$U_{ij}=\frac{n}{4}f(X_i,X_j)$$ Based on a large number of computer simulations using many nearly…
user1963
  • 545
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Finding a basis for a subspace

Find a basis for the subspace of $\mathbb{R}^4$ consisting of all vectors that satisfy $x_1 + 2x_2 - x_3 = 0$ My general idea so far is: $x_1 = -2x_2 +x_3 + 0x_4$ $x_2 = $ free $x_3 = $ free $x_4 = $ free Where should I go from here? Am I even going…
knpwrs
  • 327
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Equivalence classes of similar $2\times 2$ matrices

How can we describe the equivalence classes under the similarity relation for $2 \times 2$ matrices with respect to the field of real numbers, $\mathbb{R}$? How would the equivalence classes change if the field is $\mathbb{C}$? I know that for…
user67928
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