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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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$V$ is isomorphic to $V^{\ast\ast}$, the double dual space of $V$.

Prove that for any vector space $V$ the map sending $v$ in $V$ to (evaluation at $v$) $E_v$ in $V^{**}$ such that $E_v(\phi) = \phi(v)$ for $\phi$ in $V^*$ , is injective. Derive from this that if $\dim V < \infty$, its double dual $V^{**}$ is…
user4593
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How to prove that $f(A)$ is invertible iff $f$ is relatively prime with the minimal polynomial of $A$?

Let $A$ be a matrix from $\mathbb{M}_{n \times n}(F)$ and $f(x) \in F[x]$. How does one prove the following: $f(A)$ is invertible iff $\gcd(Ma,f)=1$ where $Ma$ is the minimal polynomial of $A$. Thanks.
user6163
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Proof for the dimension of a subspace of M(n,R)

First I will state the exercise and then show what I could get. Let $A_{1},...,A_{k} \in M(n,R)$ such that for all $ 1 \le i \neq j \le k $: $ A_{i}^{2}=I$ and $A_{i}A_{j}+A_{j}A_{i}=0$ show that $k \le \frac{n(n+1)}{2}$ Ok, first of all we can see…
Augusto Matteini
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Geometric interpretation of a vector space and subspace?

I understand how to manipulate vector spaces and subspaces and how to prove various statements about them, but I still don't fully understand what they represent geometrically. I just need an intuitive grasp as to what these are. Is a vector space a…
duncster94
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In Search of a map $\phi:\mathbf{C}\to\mathbf{C}$ satisfying additivity.

I am trying without much luck to come up with an example of a function $\phi:\mathbf{C}\to\mathbf{C}$ such that $\phi(x+y) = \phi(x)+\phi(y),\forall x,y\in\mathbf{C}$ and yet for some $a,\alpha\in\mathbf{C}$ we have $\phi(a\alpha)\neq…
atifcppprogrammer
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How to expand in a non-orthogonal basis in an intuitive way?

I have a basis that consists of four non-orthogonal vectors $\{|u_i\rangle\}, 1 \le i \le 4$. Can the formula for an orthonormal expansion be modified so that it holds true for any given basis? $$|v>=\sum_i |u_i\rangle\langle u_i|v\rangle$$ I could…
TheAverageHijano
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Let $A$ and $B$ be real matrix such that $A+iB$ is non singular show that there exist $t \in \mathbb{R}$ such that $A+tB$ is non singular

Let $A$ and $B$ be real matrices, with $A+iB$ non-singular. I need to show that there exist a real number $t$ such that $ A+tB $ is non-singular. I don't have any idea how I can approach this question... could I please get a hint?
Renu
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Natural isomorphism to dual space of an inner product space in complex case.

$k$ is a field and $V$ is a finite dimensional $k$ vector space that has an inner product $\langle - , - \rangle$. If $k = \mathbb{R}$, there is a natural isomorphism $\phi \colon V \rightarrow V^*$, $v \mapsto \langle v, - \rangle$. However, if $k…
user297841
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How to show that an $n$-dimensional vector space over field $\mathbb{F}_p$ has $p^n$ elements.

Suppose $V$ is an $n$-dimensional vector space over the finite field $\mathbb{F}_p$ for some prime $p$. How do I show that $V$ has $p^n$ elements? I was thinking that considering $n$ basis elements one can show that each element of the basis spans…
Jimmy R
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Question regarding positive-definite matrices

Let $A, B$ be positive-definite matrices and $Q$ a unitary matrix, furthermore suppose $A=BQ$. Prove or disprove: $A=B$. I'm having a hard time figuring out where to begin. Thanks.
user8577
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1 answer

Approximate a function using another function

Problem Find the best approximation of $f(t)=t^2$ with $h(t)=ae^t+be^{2t}+c$ everywhere on the interval $[0,4]$. Attempt I know how to solve this problem given sample points, by using least squares, but I am having a hard time figuring out how to…
James Mnatzaganian
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Prove that a linear combination of zero-sum vectors also sums to zero

I have a matrix $A$ whose rows sum to zero, such that $\sum_j A_{ij} = 0, \forall i$. If I multiply it by any matrix, $B$, can it be proven that the resulting matrix, $C = BA$, must also have zero sum rows? I find that they are empirically. Is such…
user664303
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Proof every operator has an upper-triangular matrix

I am having trouble understanding this proof that every operator has an upper-triangular matrix. $\lambda=$ is an eigenvalue of $T$, for $T \in L(V)$ where $V$ is a vector space on $F^n$, they say : suppose $U = \mathcal{R}(T-\lambda I)$, then…
Frank
  • 880
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unit length vector in kernel of matrix

congratulation you all passed festival(new year,christmas),guys i have question related kernel of matrix,namely suppose we have following matrix $$ A= \begin{bmatrix} 1 & -1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \\ …
dato datuashvili
  • 9,194
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Properties of a Multilinear Function

I have a question regarding the properties of a multilinear function. This is for a linear algebra class. I know that for a multilinear function, $$f(c\vec{v}_1, \vec{v}_2,\ldots,\vec{v}_n)=c \cdot f(\vec{v}_1, \vec{v}_2,\ldots,\vec{v}_n)$$ Does…
Delightful_Richard
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