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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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Prove that if $b \ne 0$ then the set of solutions to $Ax=b$ is not a subspace

I realize that I will probably have to prove that the solution set does not contain the zero vector. I've been trying to prove this, but I am not sure how to. This is what I have so far, but it doesn't sound very proofy. I'm new to proofs and…
Jason
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Why null space and column space?

I am not asking this question for WHAT is null space or WHAT is column space. I have finished learning about the definitions of these two concepts for a while. However, to install these concepts in my mind forever, I really want to know what the…
Justin Chan
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Minimal polynomial and invariant subspaces.

Let $T$ be a linear operator on the $n$ dimensional vector space $\mathbb{V}$. Suppose that $\mathbb{V} = \sum_{i=1}^{k}W_i$ where each $W_i$ is $T$ - invariant. Let $\mu_{T_i}$ be the minimal polynomial of the operator restricted to $W_i$. If the…
EuYu
  • 41,421
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Help on the relationship of a basis and a dual basis

If $B_1 = \{v_1,\ldots,v_n\}$ and $B_2 = \{v_1',\ldots,v_n'\}$ are bases for a vector space $V$, and $D_1= \{\delta v_1,\ldots, \delta v_n\}$ and $D_2 = \{\delta v_1',\ldots, \delta v_n'\}$ are the corresponding dual bases of $V^*$ prove that, if…
Pablo
  • 1,084
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How to prove $\exp(\ln M)=M$

Given a $n\times n$ real (complex) matrix $A$. Let me define: $$\exp A=\sum_{n=0}^\infty \frac{A^n}{n!}$$ and $$\ln A=\sum_{n=1}^\infty (-1)^{n+1}\frac{(A-I)^n}{n}$$ Let assume that the $2$ above series converge for $A=M$. How can I prove…
anonymous67
  • 3,458
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Proof that Every Positive Operator on V has a Unique Positive Square Root

Suppose $V$ is a finite-dimensional, nonzero, inner-product space over $\Bbb{F}$, and $\Bbb{F}$ denotes $\Bbb{R}$ or $\Bbb{C}$. My thought is : suppose $T$ is a positive operator; thus, $T$ is self-adjoint. Every self-adjoint operator on $V$ has a…
ElleryL
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Left and Right inverses of linear operators

Let $X$ and $U$ be vector spaces over a field $F$, and let $T : X \to U$. (a) If there exists an operator $S : U \to X$ such that $S(T(x)) =x$ for all $x \in X$, then $S$ is called a left inverse of $T$. (b) If there exists an operator $S : U \to…
nomadicmathematician
  • 5,511
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Changing the spectrum of a symmetric matrix by diagonal perturbations

Given a fixed symmetric matrix $S$, can one change the spectrum of $S$ to any desired set of eigenvalues $\{\lambda_1,\dots,\lambda_n\}$ by adding a diagonal matrix $D$ to $S$?
Donald
  • 191
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Proof of "Eigenvectors corresponding to different eigenvalues are linearly independent."

"Eigenvectors corresponding to different eigenvalues are linearly independent." My professor told us this during a lecture, but gave no proof or explanation.
jacob
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A square matrix has the same minimal polynomial over its base field as it has over an extension field

I think I have heard that the following is true before, but I don't know how to prove it: Let $A$ be a matrix with real entries. Then the minimal polynomial of $A$ over $\mathbb{C}$ is the same as the minimal polynomial of $A$ over…
iroiroaru
  • 93
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Linear independence of vectors over larger fields

I was just wondering whether anyone knows an answer to the following: Suppose that ${\mathbb F}$ is a subfield of a field ${\mathbb G}$ and that $v_1,\ldots ,v_k$ are linearly independent vectors in ${\mathbb F}^n$ (over $\mathbb F$). Is it…
Sean
  • 93
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Upper triangular matrix representation for a linear operator

I just the read the proof that every linear operator over finite-dimensional complex vector space has an upper triangular matrix with respect to some basis of $V$ (pretty neat fact!). I am using Linear Algebra Done Right by Sheldon Axler. My…
Prism
  • 11,162
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7 answers

Characteristic polynomial of a matrix is monic?

Given a $n \times n$ matrix A, I need to show that its characteristic polynomial, defined as $P_A (x) = det (xI-A)$ is monic. I am trying induction. But no clue after induction hypothesis.
ugstudent1243
  • 403
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Can a matrix have a null space that is equal to its column space?

I had a question in a recent assignment that asked if a $3\times 3$ matrix could have a null space equal to its column space... clearly no, by the rank+nullity theorem... but I have a hard time wrapping my head around the concept of such a matrix,…
Mirrana
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How do I prove that $x^TAy = y^TAx$ if A is symmetric?

Ok this is for a HW but I'm not looking for a handout...just a hint to get me on the right track. I have no idea where to begin proving this: Show that if A is a symmetric matrix, then $$x^TAy = y^TAx$$
GBa
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