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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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Question definition: If I take any three vectors...

I am currently studying linear algebra and one of the things I am having trouble with concerns understanding the questions being asked to me. The following question I am having trouble defining. The following question is in R2 space. If I take any…
Karl Morrison
  • 908
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Taking the Inner Product with the Zero Vector

Is it possible for the inner product of any vector with the zero vector $ \mathbf{0} $ to be nonzero? Or must it always be zero? I'm struggling to find a counterexample. That is, is the following statement correct? $$ \langle \mathbf{v},…
PolandAspect
  • 312
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Linear Algebra - eigenvalues

I have this problem: Let $A$ be any $n \times n$ matrix, defined over the real numbers, such that $A-A^2=I$. Then prove that $A$ does not have any real eigenvalues. What I did: $$A-A^2=I$$ $$A-A^2-I=0$$ $$A(A-I)-I=0$$ Now I need to show that $A(A-I)…
JaVaPG
  • 2,716
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Is support of an operator same as row space?

Support of an operator is vector space that is orthogonal to its kernel. Does this mean support is same as row space? How to calculate support for any matrix?
asdf
  • 81
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Inverse of matrix sum of identity and outer product

So before we begin, I already know the answer. I'm just having difficulty figuring out the steps for finding it. Given $u,v \in \mathbb{R}^{n}$, I want to show that $$(I+uv^{T})^{-1}= I - \frac{uv^{T}}{1+v^{T}u}$$ I know from Inverse of the sum of…
Cory Smith
  • 81
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Diagonalisable matrix

Let $A$ be a $n\times n$ matrix with complex entries. Suppose that $\operatorname{tr}(A)=\operatorname{tr}(A^{2})=\cdots \ =\operatorname{tr}(A^{n-1})=0 $, and $A^{n}\neq 0 $. Then $A$ is diagonalisable.
WLOG
  • 11,436
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Angle between two planes in four dimensions

Suppose I have two planes defined in 4D space, either in terms of vectors spanning the planes, $X = t_1 A_1 + t_2 B_2$ and $X = t_3 A_3 + t_4 B_4$ (where $X$, $A$'s, and $B$'s are vectors with four elements and $t$'s are scalars), or in terms of…
8bar
  • 115
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Power of a matrix, given its jordan form

Can someone please explain how to find the power of a matrix $A$, given $A=MJM^{-1}$ where the matrix $J$ is in the Jordan canonical form? Or else please explain how to find the powers of a matrix $J$ that is in the Jordan canonical form.
rockstar123
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Solve a linear equation system, while enforcing a unit vector solution

I have an equation system of the form Aix + Biy + Ciz = Di, where (x,y,z) is a unit vector, and (Ai, Bi, Ci, Di) are sets of measurements from a noisy system (with typically 3-5 independant readings). My first intuition to solve this problem was to…
Bossykena
  • 283
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Find all matrices satisfying $X^3=I-X$

We are studying for a qualifying exam and have come across the following problem in a previous exam. Determine the solutions (if any) of the matrix equation $X^3=I-X$ in the $2 \times 2$-matrices over $\mathbb{R}$. Any hints in the right direction…
Tyler Clark
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If $\{u,v,w\}$ is a basis for $V$, then $\{u+v+w,v+w,w\}$ is also a basis: is this proof correct?

Let $u,v,w \in V$ a vector space over a field F such that $u \neq v \neq w$. If $\{ u , v , w \}$ is a basis for $V$, then prove that $\{ u+v+w , v+w , w \}$ is also a basis for $V$. Proof: Let $u,v,w \in V$ a vector space over a field $F$ such that…
aortizmena
  • 367
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Prove that $v_1, \dots v_n$ is a basis of V.

Prove that if $e_1, \dots e_n$ is an orthonormal basis of $ V$ and $v_1, \dots , v_n$ are vectors in $ V$ such that $$\|e_j - v_j\| < \frac{1}{\sqrt{n}}$$ for each j, then $v_1, \dots v_n$ is a basis of $V$. I have fiddled around with this for a…
Soaps
  • 1,093
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Non-orthogonal projections summing to 1 in infinite-dimensional space

Consider projection operators $\rho_1,\ldots,\rho_k$ defined on vector space $V$ over field of characteristic $0$, such that $$ \rho_1+\cdots+\rho_k = 1 $$ Projections $\rho, \pi$ are said to be orthogonal, if…
Marcin Łoś
  • 2,613
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Number of solutions - please check my solution?

Determine, in accordance to $k$, how many solutions does the given system of equations have: $$ \begin{cases}kx+(k+1)y=k-1\\4x+(k+4)y=k\end{cases} $$ And check, for which values of $k$ this system has exactly one solution lying within the…
Jameson
  • 113
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If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$

How can you show that if $B$ is a maximal linearly independent set of $V$, then this implies that $B$ is a basis of $V$?
Mike Richards
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