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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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Show that $T-iI$ is invertible when $T$ is self-adjoint

Let $T$ be a self adjoint operator on a finite dimensional inner product space $V$. Then $ \| T(x)\pm ix \|^2=\| T(x) \|^2+\| x\|^2$ for all $x \in V$. Deduce that $T-iI$ is inverible. Since $\| T(x)\pm ix \|=0$ iff $\| T(x)\|=0$ and $\| x…
nonam
  • 515
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Finding minimal polynomial

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix, such that $A$ is not of the form $A=c I_n, c \in \mathbb{R}$ and $(A-2I_n)^3 (A-3I_n)^4=0$. Find the minimal polynomial of $m_A(x)$of $A$. I know that $m_A(x) | (x-2)^3(x-3)^4$, but I am…
mr.stealyourgirl
  • 563
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Linear Algebra, Parseval's Identity

How does one go about proving Parseval's identity? Let ${v_1, v_2, ..., v_n}$ be an orthonormal basis for a a finite-dimensional inner product space $V$ over some field $F$. For any $x, y$ in $V$, prove: $\langle x, y \rangle$ $=$ $\sum\limits_{i…
user79449
  • 111
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How to define linear equations in an introductory linear algebra class

I believe there is an issue of clarification with respect to the definition of linear equations in many linear algebra texts. Here is a typical one ``A linear equation in the $n$ variables $x_1,x_2, ..., x_n$ is an equation that can be written in…
Maesumi
  • 3,702
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How to show that a valid inner product on V is defined with the formula $[x, y] = \langle Ax, Ay\rangle $?

Let $A \in L(V,W)$ be an injection and $W$ an inner product space with the inner product $\langle \cdot,\cdot\rangle $. Prove that a valid inner product on $V$ is defined with the formula $[x, y] = \langle Ax, Ay\rangle $ $L(V, W)$ = The set of…
AltairAC
  • 1,114
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What are some relationships between a matrix and its transpose?

All I can think of are that If symmetric, they're equivalent If A is orthogonal, then its transpose is equivalent to its inverse. They have the same rank and determinant. Is there any relationship between their images/kernels or even eigenvalues?
google1254
  • 305
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Prove that if $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$

I have a linear algebra final tomorrow and was practicing a few proofs. I want to make sure this proof is correct. Prove that: If $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$ $$\begin{align*} A &= QR\\[0.1cm] Ax &= QRx\\[0.1cm] Ax -…
Alexander Ventura
  • 141
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Prove that if $A - A^2 = I$ then $A$ has no real eigenvalues

Given: $$ A \in M_{n\times n}(\mathbb R) \; , \; A - A^2 = I $$ Then we have to prove that $A$ does not have real eigenvalues. How do we prove such a thing?
TheNotMe
  • 4,841
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The set of all sequences of complex numbers with limit $0$ is a subspace of $\mathbb{C}^{\infty}$

In Axler's Linear Algebra Done Right, they set an example for a subspace: The set of all sequences of complex numbers with limit 0 is a subspace of $\mathbb{C}^{\infty}$, where $\mathbb{C}^{\infty}$ denotes the vector space of complex sequences over…
César Leonardo Clemente López
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What does distance of a point from line being negative signify?

When we take distance from the line, we take $$ d = \frac{ Ax_o + By_o + C}{ \sqrt{A^2 +B^2}}$$ usually with a modulus on top, now my question is if I evaluate this distance as negative what does it mean? Can I decide on which half-plane a point…
tryst with freedom
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Kernels of commuting linear operators on infinite dimensional vector space

If $S$ and $T$ are commuting operators on an infinite dimensional vector space $V$, it is in general true that $$\ker S + \ker T \subseteq \ker(ST),$$ but in general equality does not hold. A simple example is given by $S = T = \frac{d}{dx}$ on…
Max K
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Are vector spaces and their double duals in fact equal?

Excuse me, in a course of linear algebra, our assistant stated that, if $\mathbb{V}$ is a finite-dimensional vector space, and $\mathbb{W}$ its double dual, $\mathbb{V}$ and $\mathbb{W}$ are actually equal to each other; I am wondering if this has…
awllower
  • 16,536
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Algebra - Steps to rearrange a simple equation

I have the following equation: $$\frac{1000−6n}{3+10n}=k$$ I know it can be rearranged into this, but I do not quite understand how: $$(5k+3)(10n+3)=5009$$ I know the first thing I can do is multiple both sides by $3 + 10n$ to get: $$1000 - 6n = 3k…
user2924127
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Vectors Definition

What is basically a vector ? I am not a starter in learning vectors , have used them in physics and mathematics and done with 1st year of my BS in physics . But what is a vector basically ? Is it anything that has direction and magnitude ? or is it…
abc
  • 61
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Fuglede's theorem in finite-dimensional vector space

Let $V$ be a finite dimensional vector space and $A$ be normal operator on $V$ and $B$ is an operator such that $AB=BA$. Show that $BA^*=A^*B$. I guess that this problem should not be so difficult. I have tried different approaches and I got some…
RFZ
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