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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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QR factorization of complex matrix

If you have two complex numbers $a,b$ how can you find the QR factorization of $ M = \begin{bmatrix} aI_n\\ bI_n \end{bmatrix} $, I can't seem to be able to do it. I tried an earlier trick, but I now know it's not necessary true for…
simplicity
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Linear Algebra and Geometry by Kostrikin and Manin: Remark regarding diagrams and graphic representations.

On page 5 of this book there is a particular section of the book that I am having trouble trying to understand as to what the authors' are trying to point across. It is concerning linear algebra. I will place in bold the parts I need additional…
tcmtan
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What is the difference between a 2-dimensional plane in $R^3$ and $R^2$?

A $3\times 3$ matrix with $2$ independent vectors will span a $2$ dimensional plane in $\Bbb R^3$ but that plane is not $\Bbb R^2$. Is it just nomenclature or does $\Bbb R^2$ have some additional properties that other planes don't?
badmax
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Approximating commuting matrices by commuting diagonalizable matrices

Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that $A_n \rightarrow A$, $B_n \rightarrow B$. Each $A_n$ is diagonalizable and the same for each $B_n$. For every $n$, $A_n$ commutes with…
robinson
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Linear Algebra and Trig Identity Proof

I am working on the following question. It involves finding a proof for a trig identity using linear algebra. The proof is one involving $sin(\alpha +\theta)$ and $cos(\alpha +\theta)$, as you will see. I will go through where I am up to,…
Swedish-Richard
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Prove that if $A^2=0$ then $A$ is not invertible

Let $A$ be $n\times n$ matrix. Prove that if $A^2=\mathbf{0}$ then $A$ is not invertible.
Help needed
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What can we say if the dot product of two vectors is equal to 1

The question really is in the title. I know what it means if the dot product equals 0 but I find it interesting thinking what it means when it equals exactly 1 and can't seem to find anything online to enlighten me. Thanks
moony
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Eigenvalues of a rectangular matrix

I've read that the singular values of a matrix are equal to the $$\sigma=\sqrt{\lambda_{K}}$$ where $\lambda$ are the eigenvalue but I'm assuming this only applies to square matrices. How could I determine the eigenvalues of a non-square matrix.…
d1m0o
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Why do we define change of basis matrix to be the transpose of the transformation?

Example. Let $V$ be a finite dim vector space with two different bases $S = \{ u_1,u_2 \} = \{ (1,2),(3,5) \}$ and $S' = \{ v_1, v_2 \} = \{ (1,-1), (1,-2) \} $ You can check that $v_1 = -8u_1 + 3u_2$ and $v_2 = -11u_1+4u_2$ and $P =…
Lemon
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Proof of Cauchy-Schwarz inequality.

I'm trying to understand the proof of the Cauchy-Schwarz inequality: for two elements x and y of an inner product space we have $$\lvert \langle x,y\rangle\rvert \leq\lVert x \rVert \cdot\lVert y\lVert$$ The proof I am reading goes as follows: We…
Jimmy R
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Positive definiteness of a matrix

Let ${x_1,x_2,...x_n}$ be positive numbers. Consider the matrix $C$ whose $(i,j)$-th entry is $$\min\left\{\frac{x_i}{x_j},\frac{x_j}{x_i}\right\}$$ Show that $C$ is non-negative definite (or positive semidefinite, meaning $z^t C z\geq 0$ for all…
dsantanu
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Linear Algebra Done Right Exercise 3.F.26

Suppose $V$ is finite-dimensional and $\Gamma$ is a subspace of $V'$. Show that $$\Gamma = \{v\in V: \phi(v) = 0\ \forall \phi\in\Gamma\}^0$$ It's straightforward to show that $\Gamma\subset \{v\in V: \phi(v) = 0\ \forall \phi\in\Gamma\}^0$, but…
vvvvv
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How to determine if vector b is in the span of matrix A?

Given a matrix A = \begin{bmatrix} 1 &2 &3 \\ 4 &5 &6 \\ 7 &8 &9 \end{bmatrix} Determine if vector $b$ is in $span(A)$ where $$ b = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix} $$
KNgu
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Existence of T-invariant complement of T-invariant subspace when T is diagonalisable

Let $V$ be a complex linear space of dimension $n$. Let $T \in End(V)$ such that $T$ is diagonalisable. Prove that each $T$-invariant subspace $W$ of $V$ has a complementary $T$-invariant subspace $W'$ such that $V= W \oplus W'$. Note: Let…
user31899
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Giving a basis for the column space of A

Let $A = \begin{bmatrix}3&3&3\\3&5&1\\-2&4&-8\\-2&-4&0\\4&9&-1\end{bmatrix}$ Give a basis for the column space of A So what I've done so far is put it in RREF (which was a task itself) and…
Shua
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