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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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Sample points from a multivariate normal distribution using only the precision matrix?

I have a problem where I can directly compute the (sparse) precision matrix (inverse of the covariance) of a multivariate normal distribution, but the covariance itself is not sparse and I don't want to invert things. I would like to sample points…
cgreen
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What is the difference between the projection onto the column space and projection onto row space?

If I see a question that asks "find the projection a vector $b$ onto a matrix $A$" I would either solve by using $A^TA\hat x =A^Tb$ and then the projection would equal $A\hat x$, and if the matrix $A$ was orthogonal then I would use $proj_bA =…
idknuttin
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Defining the determinant of linear transformations as multilinear alternating form

Here is what our professor showed us in our linear algebra class to introduce the idea of determinants: Suppose we have an $n$-dimensional vector space $V$. Then we can create a function from $V^n$ to $\mathbb{R}$ called $vol$ (for "volume")…
Alan C
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Diagonal Matrix, just eigenvalues?

Assuming I've tested for diagonalization, can I just take the eigenvalues and arbitrarily place them in in the i,j cells to produce a diagonal matrix? Say I have a matrix $M$ with eigenvalues $\lambda_1 = 4,\; \lambda_2 = \lambda_3 = -2.$ $$M…
Cody Smith
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Can two 'different' vector spaces have the same vector?

Consider $ v_1 $ and $ v_2 $: $ \{v_1 \in \mathbb{R}^m\mid v_1 = (x_1,x_2,...,x_{m})\}\\ \{v_2 \in \mathbb{R}^{m+1}\mid v_2 = (x_1,x_2,...,x_m,0)\} $ Is $v_1 = v_2$ even they're belonging to different spaces or in this case, $ \mathbb{R}^m =…
Augusto Ferreira
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Does an eigenspace of a matrix depend continuously on its components?

Let $M(x)$ be a diagonalisable $n \times n$ complex matrix whose components are continuous functions of $x$ and suppose that, for all $x$, $M$ has eigenvalue $0$ with multiplicity $m < n$ (independent of $x$). Is it possible to choose a basis for…
octopus
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Solve for integer matrix such that $R^gu=v$ given $u$ and $v$

Given non-negative integer $n$-vectors $u$ and $v$, how does one find all $n \times n$ non-negative integer matrices $R$ and powers $g$ such that $R^gu=v$?
ecb
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Classification theorem for vector spaces

As I was going over the classification theorem for closed surfaces today, the text I'm reading gave another example of a classification theorem: finite dimensional vector spaces are classified by their dimension. As a point of fact, I think that…
user302008
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If $A$ and $B$ are normal such that $AB=0$, does it follows that $BA=0$?

If $A$ and $B$ are normal linear transformation on the finite-dimensional complex inner product space $X$ such that $AB=0$, does it follows that $BA=0$?
Silva
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Proving the intersection of distinct eigenspaces is trivial

Suppose $\lambda_1$ and $\lambda_2$ are different eigenvalues of $T$. Prove $E_{\lambda_1} \cap E_{\lambda_2}= \{\vec0\}$. I have a basic idea of what to do. Since both eigenvalues are distinct, doesn't that mean the basis for each space are…
Jay3
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Is the set $\{\frac{1}{a\,-\,\pi}\mid a\in\mathbb{Q}\}$ linearly independent over $\mathbb{Q}$?

The following problem is from Golan's linear algebra book. I have posted a proposed solution in the answers. Problem: Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Is the subset $$\left\{\frac{1}{a-\pi}\;\middle\vert\;…
Potato
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Bolzano Weierstrass theorem in a finite dimensional normed space

The problem may have a very simple answer, but it is confusing me a bit now. Let $(\mathbf{V},\lVert\cdot\rVert)$ be a finite dimensional normed vector space. A subset $\mathbf{U}$ of $\mathbf{V}$ is said to be bounded, if there is a real $M$ such…
Somabha Mukherjee
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Does there exist a vector space with 30 elements?

Does there exist a vector space with 30 elements? How to determine whether there exist any vector space of particular cardinality?
Srijan
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Why in the proof of $A\cdot Adj(A)=Det(A)\cdot I_n$ entires not on the diagonal are zero?

I have read the proof. I do not get why the entires that are not on the diagonal are equal to zero, or why the following part of the proof is true: If $k\neq \ell$ $$(A\cdot\hat A)_{k\ell}=\sum_{i=1}^n (-1)^{i+\ell}a_{ki} \det A(\ell\mid i)=0$$
gbox
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Express the polynomial $x^3-4x-4$ as a linear combination of $x-2$, $(x-2)^2$ and $(x-2)^3$

Express the polynomial $x^3-4x-4$ as a linear combination of $x-2$, $(x-2)^2$ and $(x-2)^3$ I've been looking everywhere but I still don't quite understand the question. I know that a linear combination is like a matrix consisting of a specific…
kingportable
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