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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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How to make two vectors orthogonal?

From my question on CV, I need to find a way to make my two vectors orthogonal. They are: $${\bf{v}} = \pmatrix{1 \\ 1 \\ 1 \\ 1 \\ 1 } \hspace{1.5cm} {\bf{w}} = \pmatrix{0.25 \\ 0.0625 \\ 0 \\ 0.0625 \\ 0.25}$$ What I (think) I need is the step…
Kaish
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the relationship between eigenvectors and matrix multiplication

If A has eigenvector $\mathbf{v}_1$ so that $A\mathbf{v}_1=\lambda_1\mathbf{v}_1$and B has eignenvector $\mathbf{v}_2$ so that $B\mathbf{v}_2=\lambda_2\mathbf{v}_2$, then what can you say about AB? can you say…
Vafa Khalighi
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Linear Algebra Text Problem

We have specified number of light bulbs. In addition to the array there are buttons. Pressing the button changing state of light bulbs which are connected to the switch. It is known that for each set of lamps exist button that is connected with the…
Jonny
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Solution to underdetermined linear equations

I have a set of numbers $x_i$ and I know sums of certain subsets $y_i=\sum x_{\sigma_k}$. All $x_i>0$ and I'm looking for a simple solution. With some internet research I found that this might be related to problems in signal processing. So…
Gere
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Does the zero product property hold in vector spaces?

Suppose $V$ is a vector space over a field $F$. Let $v \in V\setminus \{0\}$ and $\lambda \in F$. Does $\lambda v= 0$ imply $\lambda = 0$?
yammatack
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Linear algebra objective type question csir 2017.

For every $4\times 4$ real non singular symmetric matrix $A$ there exist a positive integer $p$ such that $pI+A$ is positive definite. $A^p$ is positive definite. $A^{-p}$ is positive definite. $ e^{pA}-I$ is positive definite. $ 1$st option is…
neelkanth
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Determine a formula for a dual basis.

Let $\beta= \{ (2,1),(3,1) \} $ be an ordered basis for $\Bbb R^2$. Suppose that the dual basis of $\beta$ is given by $\beta^*= \{f_1,f_2 \} $ To explicitly determine a formula for $f_1$ we need to consider the equations…
aortizmena
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What is happening in a linear algebra computation?

About a year ago I took a Linear Algebra class that was required for my degree. Unfortunately that class had an unidentified pre-requisite and started at a much higher level then I really needed. Going in I had no prior experience with linear…
Kenneth
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how this equation is a linear equation?

How is the equation $x_1+5x_2-\sqrt{(2x_3)} = 1$ a linear equation? The answer given in the book is, "The Equation is linear". How can an equation involving a square root like the above equation be a linear equation? here is the cutting of the…
Zia ur Rahman
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determinants of matrices of minors

Let $A$ be an $n \times n$ matrix and fix an integer $k$ with $1 \leq k \leq n$. Define a new matrix $\text{minor}_k(A)$ whose entries are the $k \times k$ minors of $A$. This new matrix will be $\binom{n}{k} \times \binom{n}{k}$. Theorem? Let $D$…
John Palmieri
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Linear algebra change of basis always makes a linear map diagonal.

Prove that there exist bases $\alpha $ and $\beta$ for V such that $ [T]_{\alpha}^{\beta} $ is a diagonal matrix with each diagonal entry equal to either 0 or 1. Originally i thought that T=I was the only solution to this i realize that is not the…
Faust
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A confusion about Ker($A$) and Ker($A^{T}$)

Today we were discussing how for an nxn orthogonal projection matrix from $\mathbb{R^{n}}$ onto a subspace W, Ker($A$)=$(Im$A$)^{\perp}$=$W^{\perp}$ and that Ker($A^{T}$) is also $W^{\perp}$. This prompted the question of what conditions are…
LinAlgStudent
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Finding two more vectors to make up an axis

Given a vector $\vec{v} = (x, y, z)$, how do I find two vectors that make up an axis with $\vec{v}$? In other words, one of them is perpendicular and lies in the same plane and the other is normal to those two vectors.
user19410
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When is product of right-invertible matrix and left-invertible matrix invertible?

I have a strong suspicion this is a textbook linear algebra problem, but I have been unsuccessful in finding an answer. Let $A$ be an $n \times m$ matrix and let $B$ be an $m \times n$ matrix where $n < m$. Suppose than $A$ has rank $n$ ($A$ has a…
John
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Let $A$ and $B$ be square real matrices such that $A+iB$ is non-singular. Show that there exists $t\in \mathbb{R}$ such that $A+tB$ is non-singular.

Let $A$ and $B$ be square real matrices such that $A+iB$ is non-singular. Show that there exists $t\in \mathbb{R}$ such that $A+tB$ is non-singular. My Attempt: I thought about considering the polynomial over $\mathbb{C}$. Let $$f(t)=|A+tB|.$$ Then…
Student
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