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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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countable dimensional vector space has uncountable eigenvalues?

Consider $\mathcal{R}^{\infty}$, and linear map $\mathcal{L} \in L(\mathcal{R}^{\infty})$, where $\mathcal{L}((x_1,x_2,...))=(x_2,x_3,...)$. Now, any number $\lambda \in \mathcal{R}$ is an eigenvalue with eigenvector $(c,c\lambda,c \lambda^2,...)$.…
user 6663629
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Geometrical meaning of the Column Space

Suppose I have $2$ planes in $R^3$ and they form a system $Ax=b$. I know the NullSpace of $A$ represents geometrically the vectors that form the intersection between the 2 planes shifted to the origin. I also know that the Row Space of A…
nerdy
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$n$ by $n$ invertible matrix $A$ has $\text{rank(A)}=n$

I'm wondering why $\text{rank}(A)=n$ means $A$ is invertible. Since invertible means one-to-one and onto, we have to prove that. $\text{rank}(A)=n$ means $\dim(N(A))=0$ which means one-to-one. Now, we have to show…
noname
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How to write the inverse matrix $A^{-1}$ as a polynomial in $A$?

I came a cross a question that gives a matrix $A$, and asks to write $A^{-1}$ as a polynomial in $A$ with real coefficients. I don't know what this means, and googling didn't clarify very much. I found the inverse $A$, but I do not know what it…
RRR
  • 321
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Determining if a Matrix is Diagonalizable without computing Eigenvalues

Is there any simple way to determine if a matrix is diagonalizable without having to compute eigenvalues? I'm motivated by the idea that for $\mathbb{R}^n$, to determine if a matrix is diagonalizable via an orthogonal transformation, you just need…
grunet
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Finding trace and determinant of linear operator

I've got the following question Consider the linear operator of left multiplication by an $m \times m$ matrix $A$ on the vector space of all $m \times m$ matrices. Determine the trace and determinant of this operator. I'm a bit stuck as to how to…
Noble.
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eigenvalues and eigenvectors for rectangular matrices

We can generalize matrix inverses from non-singular square matrices to rectangular matrices in general, for example, the well-known Moore–Penrose pseudoinverse. I am wondering how this can be done for eigenvalues and eigenvectors. Though…
Qiang Li
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Vector in 9 dimensions always has a solution?

I was watching Professor Gilbert Strang's lecture on Linear Algebra, and in the video, he talks about when we can or cannot say that linear combinations of vectors span the entire region in that dimension. He said that if we had nine vectors, each…
user681443
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What are the eigenvalues of $X = xx^{T}, x\in\mathbb{R}^{d}$?

I'm given the matrix $X = xx^{T}\in\mathbb{R}^{d \ x \ d}, x\in\mathbb{R}^{d}$. Does somebody know how to compute $\lambda_{max}(X)$ or $\lambda_{min}(X)$? I only want to know these two eigenvalues, the others are not really important. I seem to be…
Michael W.
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What is the vector $x ∈ \mathbb{R^3}$ that achieves $max||x||_1$ subject to $||x||_2 = 1$?

I'm trying to answer the questions "What is the vector $x ∈ \mathbb{R^3}$ that achieves $max||x||_1$ subject to $||x||_2 = 1$?" and "What is the vector x ∈ $R^3$ that achieves $max||x||_∞$ subject to $||x||_2 = 1$? I think the first question is…
John
  • 111
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Commuting Skew-symmetric Nilpotent 4x4 Matrices

Suppose $A$ and $B$ are nonzero, commuting, skew-symmetric, nilpotent matrices in $M_4(k)$, $k$ a field (char $k\ne 2$). Must $A=\lambda B$ for some $\lambda\in k$? I have shown that this is true for $3\times 3$ matrices, and I believe it should…
Jared
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Let $A$ be a real $n\times n$ such that the diagonal entries are positive, the off diagonal entries are negative, and the row sums are positive.

Let $A$ be a $n\times n$ matrix over the reals such that the diagonal entries are all positive, the off-diagonal entries are all negative, and the row sums are all positive. Show that $\det A \neq 0$. To show that $\det A\neq 0$, it would be…
measuresproblem
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Show that $L^nv\rightarrow 0$.

Let $V$ be a finite-dimensional normed vector space. Let $L:V\rightarrow V$ be a linear operator and let $v\in V$. Assume that there is a sequence $\{n_i\}_{i=1}^\infty\subset\mathbb{Z}$ such that $L^{n_i}v\rightarrow 0$. Show that $L^nv\rightarrow…
Q-Y
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If $\langle Ta, a\rangle \in \mathbb{R}$ for all $a$ then $T$ is self-adjoint

I have having trouble with the following question: Let $V$ be a finite-dimensional complex inner product space, and let $T$ be a linear operator on $V$. Prove that if $\langle T\alpha, \alpha\rangle$ is real for every $\alpha$ in $V$ then $T$ is…
providence
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Null spaces and invertible matrix

If $A$ and $B$ are $n×n$ matrices, show that they have the same null space if and only if $A = UB$ for some invertible matrix $U$. I started the question by saying $Ax = 0$ for some vector $x$ in $\text {null}(A)$. Now I'm lost. Could someone…
John Lee
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