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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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Prove: Every Subspace is the Kernel of a Linear Map between Vector Spaces

I'm trying to show that, given two finite-dimensional vector spaces $V,W$, and any subspace $V'$ of $V$, that there is a linear map $T:V\to W$, whose kernel is precisely $V'$, given the condition that $\dim V-\dim(\ker T)<\dim W$. I would like to…
Jay K
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Division of a matrix polynomial

I'm trying to show: Let $C(t)=C_rt^r+C_{r-1}t^{r-1}+\cdots+C_1t^1+C_0\in \mathcal{M}_n(\mathbb{F}[t])$ a polynomial with coefficients $C_{i}$ in $\mathcal{M}_n(\mathbb{F})$. Show that, exists a matrix polynomial $Q(t)$ such…
Hiperion
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Prove that for any positive integer $n$, $A^n ≠ I$.

Let $A$ be a $2\times 2$ matrix with $tr(A) > 2$. Prove that for any positive integer $n$, $A^n ≠ I$. I feel like I should approach this with respect to eigenvalues, i.e. the sum of the eigenvalues of $A$ is greater than $2$. However, I don't know…
linearalgebrathrowaway
  • 315
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To find jordan canonical form

Which of the following matrices have Jordan canonical form of equal to the $3\times 3$ matrix $$ \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ a)$ \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 &…
Tani
  • 1,026
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A question about biorthogonal basis composed of eigenvectors of a finite-dimensional non-self-adjoint matrix

The non-self-adjoint matrix M has non-degenerate eigenvalues, that is $M \psi_i = e_i \psi_i$, and its adjoint matrix satisfies $M^\dagger \chi_j= e_j^* \chi_j$. I know that $(\chi_j, \psi_i) = (\psi_i, \chi_j ) = 0$ for $i \neq j$, but why…
user234999
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Is the sum of two projections a projection?

Let $ S $ and $ T $ be two linear subspaces of $ \Bbb{R}^{2} $. Then is the sum of the projections $ P_{S} $ and $ P_{T} $ (i.e., $ P_{S} + P_{T} $) a projection? I don’t think it is since the projection rule doesn’t hold ($ P_{M}^{2} $ does not…
ann
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Prove that every elementary matrix is invertible, and the inverse is again an elementary matrix.

I know that there are many proofs regarding this. However, the book i'm using seems to suggest another way to do it without giving an answer. What i mean by the another way is some other proofs that do not use the fact that elementary row operation…
241324792581
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Under what conditions is the product of two invertible diagonalizable matrices diagonalizable?

The answer in this question gives an example for the statement product of two invertible diagonalizable matrices is not diagonalizable. My question is: Are there some conditions, perhaps on the eigenvalues and eigenvectors of matrices, under…
user166467
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operation to get a diagonal matrix from a vector

In many programs you can create diagonal matrix from a vector, like diag function in Matlab and DiagonalMatrix function in Mathematica. I'm wondering whether we can use matrix product (or hadamard product, kronecker product, etc) of a vector and…
iridium
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A "wrong" counterexample to splitting lemma in linear algebra

I have a "wrong" counterexample to the following statement in linear algebra but I don't see why it's wrong: let $T:V\to W$ be a linear map between vector spaces. Then, $V$ is the direct sum of $\textrm{im}(T)$ and $\ker(T)$. Let $V$ be the space of…
Vladimir
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show that $\det{(A^2+B^2+C^2-BA-CB-AC)}=0$

Let $A,B,C\in M_{2}(C)$ such that $$A^2+B^2+C^2=AB+BC+CA$$ show that $$\det{(A^2+B^2+C^2-BA-CB-AC)}=0$$ from:matrix indentity
lenovo links
  • 99
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Unitary and Upper Triangular Matrix

I am trying to prove that a matrix that is both "unitary and upper triangular" must be a diagonal matrix. I am thinking that the fact that columns of all unitary matrices form an orthonormal basis of F^n will ensure that all columns of this matrix…
Tav
  • 527
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Matrix multiplication by scalar is commutative

Is matrix multiplication by scalar commutative, i.e. $(\alpha M)N=M(\alpha N)$? If so, can we prove it without induction?
user4205580
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Question about complete orthonormal basis

Let $V$ be an inner product space. Let $W$ be the Hilbert space obtained as the completion of $V$. Is there a complete orthonormal basis of $V$ which is still complete in $W$? This is true if we assume that $V$ is separable (Schumidt's method), but…
Nancy
  • 51
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Inner Product on $\mathbb{R}$ and on $\mathbb{C}$

This is a question of the book Linear Algebra of Kenneth Hoffman. Describe explicitly all inner products on $\mathbb{R}$ and on $\mathbb{C}$. I think that if $<,>$ is one inner product on $\mathbb{R}$ (or $\mathbb{C}$) then $< ,>=K[,]$, where $[,]$…
Pitágoras
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