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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Adjoint of a linear transformation in an infinite dimension inner product space

We learned that if $V$ is a finite inner product space then for every linear transformation $T:V\to V$, there exists a unique linear transformation $T^*:V\to V$ such that $\forall u, v \in V: (Tv, u)=(v, T^*u)$. The construction of $T^*$ used the…
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Is the inverse of a linear transformation linear as well?

A question from the field of Linear Algebra. If I have a linear transformation $T$ that is one-to-one and onto, would that mean that the $T^{-1}$ will also be linear? If so, is there any general proof for it? Thanks
vondip
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Degrees of freedom for a matrix

What does it mean for a matrix to have degrees of freedom? How does the degrees of freedom relate to constraints on what those values could be in the context of an optimization problem? I'm specifically confused about the last paragraph in this…
Carpetfizz
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What is the difference between linear space and a subspace?

If W is a subspace, is it also a linear space? If V is a linear space, is it also a subspace? I am having trouble wrapping my head around the difference between the two, as it seems that the way the book defines them is the following: both have to…
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Proving the Product of Unitary Matrices is also Unitary

This is my attempt so far: Suppose I have two unitary matrices, A and B, such that $A^*A=I$ and $B^*B=I$. We want to show that $(AB)^*(AB)=I$. So we have that $$(AB)^*(AB)=B^*A^*AB=B^*IB=I.$$ Does this work as a proof? Do I need to show anything…
Walter
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If $A+A^T$ is negative definite, then the eigenvalues of $A$ have negative real parts?

Assume that $A$ is non-symmetric. "If $A+A^T$ is positive definite, the eigenvalues of $A$ have positive real parts" (originally, it was with negative definite/negative real parts, but I suspect it doesn't matter). This was claimed in a comment of…
jkn
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Matrices with real entries such that $(I -(AB-BA))^{n}=0$

I was just trying out some problems, when i couldn't solve this question: Does there exist $n \times n$ matrices $A$ and $B$ with real entries such that $$ \Bigl(I - (AB-BA)\Bigr)^{n}=0?$$ I really don't know how to proceed for this question. What…
anonymous
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Why is it called "Orthogonal Projection"? Why not just "Projection"?

Right now, we are learning decomposing vectors, but something I don't understand is the names given to this stuff For instance, in the text, the parallel component of y is said to be the orthogonal projection of y onto u. This makes no sense to me.…
mim
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If $A^3=A+I$, then $\det A>0$

If $A$ is a $n\times n$ matrix such that $A^3=A+I$, then $\det A>0$. I don't know how to solve this problem. It's easy to see that $\det A\neq 0$. Suppose $\det A<0$. These are ways that I tried: $\bullet$ $\det A^3=(\det A)^3<0$, thus…
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What is the difference between Eigenvectors and the Kernel or Null Space of a matrix?

I am just wondering what is the difference between Eigenvectors and the Kernel or Null Space of a matrix? The kernel for matrix A is x where, Ax = 0 Isn't that what Eigenvectors are too?
User
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How to determine if a linear system is solvable

I have this problem $$\begin{array}{rcccccl} 3x &-&y &+& 2z &=& 2\\ 2x &+& y &+& z &=& -1\\ x &+& 3y & & &=&-1 \end{array}$$ This gives me the matrix: $$\left(\begin{array}{rrr|r} 3 & -1 & 2 & 2\\ 2 & 1 & 1 & -1\\ 1 & 3 & 0 &…
brandon
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Characteristic polynomial of a nilpotent matrix

Let $A$ be $n\times n$ nilpotent matrix. How to calculate its characteristic polynomial? I know it should be $X^n$, but I don't know why?
Mohan
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Limit of a matrix multiplication

How can I calculate this limit: $\displaystyle\lim_{n\to\infty}\begin{bmatrix}0.9 & 0.2\\0.1 & 0.8\end{bmatrix}^n$ What is the tool that i need to aply? eigenvalues and eigenvectors? diagonalization? canonical form? (This came in a contest and was…
Julio
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$SAS^{-1}=\lambda A$ - show $\lambda^n=1$or A is nilpotent

I found a question im struggling with in http://www.math.upenn.edu/ugrad/calc/m240/240la.pdf Consider an $n*n$ matrix A with real or complex coefficients and $S$ an invertible matrix. show that if $SAS^{-1}=\lambda A$ then $\lambda^n=1\ or\ A\ is\…
Daniel s
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Orthogonal basis for infinite-dimensional vector spaces

It is quite simple to show that all finite-dimensional vector spaces with inner product have an orthogonal basis, with the standard definition of a basis from linear algebra. However, I am in trouble to find any reference about infinite-dimensional…
Ketty
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