Mathematics Stack Exchange
Mathematics Stack Exchange

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
9
votes
3 answers

Do complex eigenvalues of a real matrix imply a rotation-dilation?

This is part of a bigger proof that if there is a compact set, $K \subset \mathbb R^n$ such that the linear transformation $L$ maps $K$ into its interior, the eigenvalues $\lambda_i$ are all of absolute value less than 1. Clearly, if $L(K)\subset…
Steven-Owen
  • 5,556
9
votes
1 answer

Construction of Free Vector Spaces

I've been studying the construction of Free Vector Spaces and I want to confirm if my conclusions are correct. Given a set $A$ we wish to construct a vector space $F(A)$ which intuitively is the vector space whose elements are linear combination of…
Gold
  • 26,547
9
votes
4 answers

Invariant Subspace of Two Linear Involutions

I'd love some help with this practice qualifier problem: If $A$ and $B$ are two linear operators on a finite dimensional complex vector space $V$ such that $A^2=B^2=I$ then show that $V$ has a one or two dimensional subspace invariant under $A$ and…
Jonathan Beardsley
  • 4,503
9
votes
2 answers

formula for square of absolute value of difference of two variables $|a-b|^{2}$

Is there any formula for the expansion of $|a-b|^{2}$ ? Can I expand it in the same way as $(a-b)^{2}$?
user13267
  • 365
9
votes
3 answers

Why the transpose of a singular matrix is singular?

I have to prove this lemma without using the concept of rank neither the concept of determinant: $A$ is a singular matrix iff $A^T$ is singular Unfortunately i've only found proofs that contains rank and determinant. Can you help me ?
Koinos
  • 191
9
votes
1 answer

Do $T$-invariant subspaces necessarily have a $T$-invariant complement?

Suppose $T$ is a linear operator on some vector space $V$, and suppose $U$ is a $T$-invariant subspace of $V$. Does there necessarily exist a complement (a subspace $U^c$ such that $V=U\oplus U^c$) in $V$ which is also $T$-invariant? I'm curious…
Hailie Mathieson
  • 1,950
9
votes
3 answers

Prove: if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.

From Exercise 6 of Sec 1.2 of Linear Algebra by K.Hoffman and R.Kunze. Equivalence is defined as follows: Two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations in the other …
user470904
  • 729
9
votes
2 answers

generalizations of determinant and trace

There are $n$ symmetric polynomials in the eigenvalues of a square matrix. Two of these are the determinant and the trace, each of which have countless applications and interpretations in algebra and geometry. What about the other symmetric…
user7530
  • 49,280
9
votes
3 answers

Show that $|A|$ is an integer multiple of 125

Let A=$(a_{ij})\in M_{4\times4}(\mathbb Q)$ be a matrix each entry of which is either -2 or 3. Show that $|A|$ is an integer multiple of 125. Would it be best to look at the $2\times 2$, minor determinants, or is there some better way?
somethin'
  • 227
  • 1
  • 5
9
votes
3 answers

Why does the column space of a linear transformation equal its image?

I'm having trouble understanding this. Why does the column space of the matrix of a linear transformation equal the image of the linear transformation?
user45566
9
votes
1 answer

How to convert a random matrix to Unitary Matrix?

I know that a complex matrix $n \times n$ is said to be unitary if $AA^*=A^*A=I$ or equivalently if $A^*=A^{-1}$. But I asked what if there is a random matrix and we want to turn it into an unitary matrix, please also give an example.
Nadia
  • 91
9
votes
2 answers

If a matrix is invertible, is its multiplication commutative?

The question is prompted by change of basis problems -- the book keeps multiplying the bases by matrix $S$ from the left in order to keep subscripts nice and obviously matching, but in examples bases are multiplied by $S$ (the change of basis…
LinAlgStudent
  • 2,039
9
votes
2 answers

Intuition behind Jacobian of the SVD

I'm having a little trouble understanding the meaning behind the Jacobian of an SVD. I understand what the Jacobian is, but I don't see how you can derive a Jacobian from the SVD. To me, the SVD is just USV_transpose - I don't see how a matrix can…
Jason Cho
  • 91
9
votes
3 answers

Why is QR factorization useful and important?

Why do we need QR factorization? Is this used in any particular field?
matqkks
  • 2,860
  • 3
  • 28
  • 36
9
votes
3 answers

linear transformation $T$ such that $TS = ST$

Let $V$ be a finite-dimensional vector space over $F$. Let $T:V \rightarrow V$ be a linear transformation such that $ST=TS$ for all linear transformations $S:V \rightarrow V$. Show that $T = cI_v$ for some $c \in F$.
apple mcdonald
  • 249
  • 2
  • 4
1 2 3
…
99 100