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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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The group of invertible matrices with entries in Z

Let $G=GL_{n}(\mathbb{Z})$, the group of invertible matrices with entries in $\mathbb{Z}$. Then show that $G=\{A\in M_n(\mathbb{Z}) : det(A)=1 \,\,\,or -1\}$ Can you help me please?
user114952
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How do i prove that every linear operator between finite-dimensional Hilbert spaces is bounded?

When I learned basic linear-algebra, "adjoint" was only defined for linear operator between finite-dimensional inner product spaces. Right now, I'm studying Hilbert spaces and I want the past definition consistent with a new definition. I have…
Mathems
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Is the matrix $R$ in the $QR$ decomposition unique?

I'd like to know for positive diagonal elements why is $R$ in $QR$ decomposition unique. My guess is it must have something to do with linearly independence of the column of $R$, but then I can think of a property that lead me to uniqueness of $R$.…
Gigili
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Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular.

I'm trying to prove that statement: Prove that $\lambda = 0$ is an eigenvalue if and only if $A$ is singular. I'm not sure if my proof is totally correct: Suppose that $\lambda = 0$ if det(A) = $\lambda_1 \cdot \lambda_2 . . .\lambda_n = 0$ then A…
Mark
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Let $A$ be a $2 × 2$ matrix with real entries which is not a diagonal matrix and which satisfies $A^3 = I$. Pick out the true statements:

Let $A$ be a $2 \times 2$ matrix with real entries which is not a diagonal matrix and which satisfies $A^3 = \mathcal{I}_2$. Pick out the true statements: $\operatorname{tr}(A) = −1$ $A$ is diagonalizable over $\mathbb{R}$ $λ = 1$ is an eigenvalue…
chopak
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A Householder matrix is symmetric

I want to show that a Householder matrix is symmetric, so I must show that $H^T = H$, but from the formula $$H= I - (uu^T/\beta),$$ they are not equal. What's wrong with my reasoning? EDIT: I forgot that $(uu^T)^T$ would be $(u^T)^T(u)^T$ from the…
Gigili
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How to calculate this determinant of matrix raised to high power?

I'm trying to get ready to my exam from linear algebra by doing some random tasks and with this one i'm pretty stuck. $$A = \begin{pmatrix} 8-5i & -6 \\ 4-5i & -3+5i \end{pmatrix}$$ Given this matrix $A$ i have to calculate determinant of…
Krzysztof Lewko
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If a matrix commutes with all diagonal matrices, must the matrix itself be diagonal?

I'm new to stackexchange so feel free to correct my style/format/logic etc. The question is this: let's say $A$ is a square matrix of size $n$. I would like to show that $AD = DA$, for any diagonal matrix $D$ also of size $n$, if and only if $A$ is…
user118399
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Finding a basis to a vector space

Let $ W = \left \{\mathbf{x} = \begin{pmatrix} x_1 \\x_2 \\x_3 \end{pmatrix} : x_1 + x_2 + x_3 = 0 \right\}$ and find a basis for $W$ I don't really know how to do it by guess work so I tried this method: Solve $x_1 + x_2 + x_3 = 0$ to row echelon…
user113494
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Writing an invertible $2\times2$ matrix as a conjugate of an upper triangular matrix

It's been a while since I've studied linear algebra, and I wanted to follow up on something I read on MathOverflow. In this answer, KConrad mentions you can write any invertible $2\times2$ matrix as a conjugate of an upper triangular matrix. How…
Gotye
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Finding the basis, difference between row space and column space

I'm confused in Linear Algebra when finding the basis. In my textbook there are two methods: Row space and Casting out In the Row Space algorithm I form the Matrix whose rows are the given vectors, then I reduce it to echelon and my basis are the…
user115919
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How Find $\cos{(\pi A)}$ if $A$ is Orthogonal matrix

let $A_{n\times n}$ is Orthogonal matrix, Find the value $$\cos{(\pi A)}=?$$ and before I guess $$\cos{(\pi A)}=E-2A$$ Now this is wrong,and this problem relsut is what? I know this http://en.wikipedia.org/wiki/Matrix_exponential and we konw if…
user94270
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Is this function surjective; infinite linear combination.

Let $A(t):[0,T] \rightarrow \mathbb{R^{n\times m}}$ be continuous function. Let $$U = \text{span}\left( \bigcup_{t\in [0,T]} \text{span}(A(t)) \right)$$ Define function $f(u): L^\infty ([0,T],\mathbb{R}^m) \rightarrow U$ by: $$f(u) = \int_0^T…
tom
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Prove that every hyperplane is a null space of a linear functional

How can we prove that every hyperplane (a subspace of dimension $n-1$) is a null space of a linear functional? I don't know how to prove this. I tried a lot, but something is missing.
aiswarya
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Show that $(1,x,x^2),(1,y,y^2),(1,z,z^2)$ form a basis of $\mathbb{R}^3$ iff $x\neq y, x \neq z, y \neq z$

I'm having some trouble with this one because I always get negated statements. If I try to prove both direction directly I get that three elements are all not equal to each other and the three vectors form a basis, or if I proof by contrapositive I…
eager2learn
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