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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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What is meant by linearity of a dot product?

I would like to know what is meant by linearity of a dot product. Thank you
user99069
  • 63
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What are we solving with row reduction?

Why are row reductions so useful in linear algebra? It is easy to get lost in the mechanical solving of equations. I know I can get a matrix into reduced row echelon form. But what are the outcomes of this? What can this mean?
Rook
  • 211
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if $A^2 \in M_{3}(\mathbb{R})$ is diagonalizable then so is $A$

Prove or disprove: if $A^2 \in M_{3}(\mathbb{R})$ is diagonalizable then so is $A$. I'm pretty confident this is not true, but I've tried and tried to find a counter example without success. If someone contradicts this, I'd appreciate if you can…
daniel.jackson
  • 2,361
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Define constant c so an equation system either has none, one or infinite solutions

I have the following equation system matrix: $$\left[\begin{array}{ccc|c} c & 1 & 1 & 1 \\ 1 & c & 1 & 1 \\ 1 & 1 & c & 1 \end{array}\right]$$ From this one I'm supposed to be able to define the constant $c$, for creating an equation with…
user93603
6
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2 answers

Proof that a linear transformation is isomorphic

As homework, I need to proove whether a few linear transformations are isomorphic or not, however i do not know how to achieve this. First of all i have proven that the following map is linear: $$f:\mathbb{R}^2\mapsto\mathbb{R}^2, f\left(…
Christian Ivicevic
  • 3,123
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Given a linear map $T:V\to V$, is it true that $V=\ker(T) \oplus \mathrm{im}(T)$?

I was wondering if $V=\ker(T) \oplus \mathrm{im}(T)$ if $T:V \to V$. I know the theorem that if $T:V\to W$ is linear then $\dim(V) = \dim(\ker(T)) + \dim(\mathrm{im}(T))$. This should imply $V=\ker(T) \oplus \mathrm{im}(T)$ because if $\dim(V) =…
blue
  • 2,884
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Prove that the determinant of this matrix can be represented as a polynomial.

Let $A=(A_{ij})$ be a square matrix of order $n$. Verify that the determinant of the matrix $\left( \begin{array}{ccc} a_{11}+x & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22}+x & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2}…
Twnk
  • 2,436
6
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2 answers

Determinant of the product equal to the product of determinants?

Let $X$ be an $n\times p$ matrix and $A$ be a $n\times n$ matrix. When is it true that $$\det (X^{\top}AX) = \det(A)\det(X^{\top}X)?$$
Dexter
  • 63
6
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3 answers

Linear algebra: rank

Let $A:E\rightarrow F$ be a Linear Transformation between finite dimensional vector spaces, with $\mathrm{Rank}(A)=r$ and $\dim E=n$, $\dim F=m$. Prove that there are basis in $E$ and $F$ such that the matrix of $A$ has $a_{11}=\cdots=a_{rr}=1$ and…
Ivan3.14
  • 747
6
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Geometric interpretation of a non-symmetric matrix having only real eigenvalues?

Is there a geometric interpretation of a non-symmetric matrix having only real eigenvalues? It appears that multiplying random matrices with IID random entries eventually produces a matrix with only real eigenvalues, wondering if this can be turned…
Yaroslav Bulatov
  • 5,327
6
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2 answers

Minimal number of multiplications required to invert a 4x4 matrix

Four by four real (well, floating point...) matrices are used in computer graphics to represent projections. Sometimes we need to compute their inverses. How many multiplications are required? Valve's Source SDK implements the "pen&paper"…
Jacopo Notarstefano
  • 2,297
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3 answers

Real Symmetric Matrices

Let $A$ be a real, symmetric, $n$ x $n$ matrix. Suppose $A^m=I$ for some $m$. Prove $A^2=I$. I think I want to use the symmetric implies diagonalizability...and take powers from there...correct?
Johnny Apple
  • 4,211
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Proving that $\det{(A^i_j})= \sqrt{ |\det{(G)}|}$

Let $V$ be an $n$-dimensional vector space and let $(v_1, \dots, v_n)$ denote any oriented basis for $V$. Also, let $g$ be an inner product on $V$ and let $G$ denote the Gram matrix of inner products $G = [g(v_i, v_j)]$. I am trying to show that if…
ItsNotObvious
  • 10,883
6
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1 answer

Interpreting a singular value in a specific problem

In a similar spirit to this post, I pose the following: Contextual Problem A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that…
Paul
  • 2,133
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1 answer

How to determine if a subspace of $ \mathbb{R}^n $ has an integer basis

Let $ W $ be a sub vector space of $ \mathbb{R}^n $. How can we determine if $ W $ admits an integer basis? This is equivalent to asking how to determine if $ W \cap \mathbb{Z}^n $ spans $ W $. Obviously if $ W=\mathbb{R}^n $ then there is always an…
Ian Gershon Teixeira
  • 8,010
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