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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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Projections onto ranges/subspaces

I'm stuck on a review problem. Consider the matrix: $$\left[ \begin{array}{ccc} -1 & 1 \\ 1 & 1\\ 2 & 1 \end{array} \right] $$ I'm asked to find a matrix $P$ which projects onto the range of $A$, with respect to the standard basis. I'm not…
John Doe
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Is the dot product valid for coordinate vectors of infinite length?

On Wikipedia, it says the dot product is valid for "any number of dimensions." Let's call it $n$. $$u\cdot v = |u||v|\cos(\theta)$$ Is this still true if we let $n$ go to infinity? EDIT: By coordinate vectors of infinte length, I mean coordinate…
transcendental
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Adding a positive semidefinite matrix to a square matrix

Can one find some square matrix $A$ and a square positive semidefinite matrix $B$, such that the largest eigenvalue of $C=A+B$ is smaller than the largest eigenvalue of $A$?
danielrch
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Reference for Generalized Eigenvectors

I am looking for references on generalized eigenvectors and Jordan matrix representation. I would like a brief but complete introduction of this concepts with a nice treatment of the most important properties. I'm studying automorphisms of the torus…
Jarana
  • 705
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Practical Application of Linear Transformation as taught in Linear Algebra.

Can anyone provide a basic practical use of linear transformation? ( Or maybe a metaphor like example) Also visual, practical, or real applications for one to one onto null(A) determinant rank I'm a person who likes to picture big, and it's very…
electronicsSS
  • 101
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Questioning the need of Dual Space

I am dealing with the dual spaces for the first time. I just wanted to ask is their any practical application of Dual space or is it just some random mathematical thing? If there is, please give a few.
Manish
  • 565
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4 answers

Prove that the polynomial $q$ exists.

Suppose $p \in P(R)$. Prove that there exists a polynomial $q \in P(R)$ such that $5q''+3q' = p.$ I have already proved this using algebra, does anyone know how to do this using linear algebra. Would I have to show that p is in the span of $q''$ and…
Soaps
  • 1,093
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Help to understand Gram-Schmidt Proof

This is from Axler's Linear Algebra Done Right 6.20 proof I don't understand how does Axler get to equation 6.23. It seems to me that he simply add a vector $v_j$ such that $v_j$ is orthogonal to every vector in $\mathrm{span}(e_1,\:...\:e_{j-1})$…
ElleryL
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Compute eigenvalues

If we know that $A$ is a symmetric $2$ by $2$ matrix, $\displaystyle \max_{\|x\|=1}\:x^{T}Ax=a$, and $\displaystyle\min_{\|x\|=1}\:x^{T}Ax=b$ for some given $a, b$, can we compute the eigenvalues of $A$?
LJR
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Basis change for gradient approximation

This is specifically a linear algebra question, but I kind of need to explain context. I suspect that this is simply a basis change problem, but I'm not entirely sure. Also, please feel free to correct any glaring mistakes I make. The image below…
jhc
  • 185
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2 answers

$A^\mathrm{T}A=B^\mathrm{T}B \Leftrightarrow \exists$ orthogonal $Q$ such that $A=QB$?

Assume $A,B \in \mathbb{R}^{m\times n}$, how can you prove the following: $A^\mathrm{T}A=B^\mathrm{T}B \Leftrightarrow \exists$ orthogonal $Q$ such that $A=QB$ or is there a counterexample? Intuitively it makes sense to me, but I haven't found a…
Dimitar Ho
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3 answers

Where are linear equations with large number of variables used?

Do weather prediction / financial models or missile / rocket trajectory prediction model use these equations? What method or algorithm is used for the same?
scorpian
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A basis for the dual space of $V$

We let $\mathcal{B} = \{\alpha_1, ... \alpha_n\}$ be a basis for $V$, a vector space with inner product $\langle \cdot, \cdot \rangle$. Then we define $f_i(v) := \langle v, \alpha_i \rangle$. Show that $f_1, ..., f_n$ is a basis for $V^{\star}$ the…
Deven Ware
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3 answers

positive definiteness similarity

Does the concept of matrix similarity apply to the condition < Xv,v >? In other words, if a real square non-symmetric matrix X is similar to a symmetric positive definite matrix, do we have < Xv,v > > 0 for all nonzero vector v? I feel this is a…
Lio
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Prove that the determinant of a transformation matrix does not depend on the basis

Let $T: \Bbb U \to \Bbb V$ be a linear transformation representable by $A \mathbf x$ in some basis $B$, where $A$ is a matrix and $\mathbf x$ is a member of $\Bbb U$. $\ $Show det(A) does not depend on the basis chosen.
user156118
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