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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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Why is rotation about the y axis in $\mathbb{R^3}$ different from rotation about the x and y axis.

In my textbook for a counterclockwise rotation about the x-axis we have $\begin{pmatrix} 1 & 0 & 0\\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}$ For rotation about the z-axis we have $\begin{pmatrix} \cos\theta &…
Craig
  • 652
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Which matrices are conjugate to an integer valued matrix?

If I have a matrix $A \in M_{n\times n}(\mathbb{C})$, when does there exist a change of basis $B \in Gl_n(\mathbb{C})$ so that $BAB^{-1} \in M_{n\times n}(\mathbb{Z})$? Case $n=1$ is obvious (in this case, $A$ is a number, so $A$ must be an…
GraduateStudent
  • 881
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Dual basis with respect to bilinear form

Let $V, W$ be $n$-dimensional $K$-vector spaces with a non-degenerate bilinear form $(\cdot, \cdot) : V \times W \to K$. We call a basis $(\beta_1, \dots, \beta_n)$ for $W$ dual to a basis $(\alpha_1, \dots, \alpha_n)$ for $V$ with respect to…
Mark
  • 1,433
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Left and right multiplying of matrices

I am new to matrix multiplication and trying to understand something. Suppose you have a matrix equation $ A x=b $. I know to solve for $x$ you should left multiply by the inverse of A. But what is the reason you can't solve for $b$ like this: $ A…
Anna
  • 101
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Basis-free formulation of Jordan normal form theorem

Is there a basis-free formulation of Jordan normal form theorem? From some search I did in Google, the answer is apparently yes. But I didn't find any article that I could understand. (I've only taken two semester course in linear algebra.) My…
makela
  • 101
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annihilator linear algebra

I have a problem as follows: $W_1$ and $W_2$ are subspaces of a finite-dimensional vector space $V$. $W^0$ is the annihilator of $W$. (a) Prove $(W_1 + W_2)^0 = W_1^0 \cap W_2^0$. (b) Prove $(W_1 \cap W_2)^0 = W_1^0 + W_2^0$. Thoughts so far: By…
Megan
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eigenvalues of a matrix $A$ plus $cI$ for some constant $c$

If $A$ is a $n \times n$ real matrix with eigenvalues $\lambda_1,\lambda_2,...\lambda_n$, how does one get the eigenvalues of the matrix $A$ + c$I$, where $I$ is the identity matrix and $c$ is a non-zero real constant? I tried to work out the…
user74261
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When does a matrix have a cyclic decomposition and when a rational form?

I'm reading Hoffman's "Linear Algebra" and this question comes to my mind: suppose $V$ is a finite-dimensional vector space on a field $F$, and $T$ a linear operator on $V$, when does a matrix has Cyclic Decomposition, and when does it has a…
athos
  • 5,177
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What does a positive definite matrix have to do with Cauchy-Schwarz Inequality?

In my text book, Cauchy-Schwarz Inequality is extended to a positive definite matrix. But I neither understand what the relationship between Cauchy-Schwarz Inequality and a positive definite matrix nor the sentence underlined in red, I am not a…
user122358
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4 answers

If $A^TA$ is invertible, then $A$ has linearly independent column vectors

Question: Prove that for a $m \times n$ matrix $A$, if $A^TA$ is invertible, then $A$ has linearly independent column vectors. I am hitting a complete blank with this proof, I have the following jotted down so far about stuff that I know. What I…
user860374
  • 4,131
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How to prove the Pythagoras theorem using vectors

I've got a question concerning how to proof the Pythagoras theorem using the following assumption: $x$ is perpendicular to $y$ (if and only if) $||x+y||^2 = ||x||^2 + ||y||^2$, where $x$ and $y$ are vectors. I have a basic understanding of linear…
Rope
  • 103
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$T$ is diagonalizable if $T^n$ is identity for some $n$

Suppose $T$ is a linear operator on a $\mathbb{C}$-vector space $V$. Further, assume $T^n$ is the identity operator, for some $n$. Then, $T$ is diagonalizable. I think there is a proof using Jordan theory, but I wish to find one without using it.
saubhik
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"All positive solution" to system of linear equations

My question is: Are there any criteria to decide whether a system of linear equations allows a solution where all variables are greater than 0? Clearly, I could compute the solution space and check if there is a solution that satisfies the…
Oliver Witt
  • 201
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6 answers

Apostol proof divides by zero?

The following is a proof from Apostol's linear algebra book that $\{1,t,t^{2},...\}$ is independent. To my eye, he's dividing by zero repeatedly. Is this really as huge an error as it seems, or are there missing details that would make this…
mbenoni7
  • 103
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How to prove there exists a linear transformation?

I've been given the following problem as homework: Prove that there exists a linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ such that $T(1,1) = (1,0,2)$ and $T(2,3) = (1,-1,4)$. Since it just says prove that one exists, I'm guessing I'm…
Casey Patton
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