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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
22
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Is there an elegant way to prove a function is linear?

I'm reading Hoffman and Kunze's linear algebra book and on page 73 in the exercise 7, they ask to verify this function $$T(x_1,x_2,x_3)=(x_1-x_2+2x_3,2x_1+x_2,-x_1-2x_2+2x_3)$$ is a linear transformation. This exercise is really simple, but a…
user42912
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21
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2 answers

Does the set of matrix commutators form a subspace?

The following is an interesting problem from Linear Algebra 2nd Ed - Hoffman & Kunze (3.5 Q17). Let $W$ be the subspace spanned by the commutators of $M_{n\times n}\left(F\right)$: $$C=\left[A, B\right] = AB-BA$$ Prove that $W$ is exactly the…
EuYu
  • 41,421
21
votes
3 answers

Adjoint of derivative operator on polynomial space

I was working on a problem when I made the following reasoning. I know that every linear operator $T:V \longrightarrow V$ on a Hilbert space $(V,\langle.,.\rangle)$ such that $\dim(V)<\infty$ has one (unique) adjoint operator $T^*:V \longrightarrow…
Wheepy
  • 721
21
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3 answers

If $A$ and $B$ are positive definite, then is $B^{-1} - A^{-1}$ positive semidefinite?

I've found this while googling some properties of positive semidefinite matrices. (Unfortunately, I cannot remember where I've discovered it.) If this is true, it'll greatly save my time in my work. Is it true? How can you prove it? Let's say I…
user19906
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21
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2 answers

Prove that vector space and dual space have same dimension

As an exercise in my textbook, I need to prove that if $V$ is a finite dimensional vector space with dual space $V^*$ over $\mathbb{R}$, then dim$(V)$=dim$(V^*)$. Let $\omega\in V^*$ and let $\{e_1,...,e_n\}$ be a basis for $V$. Define $e^i\in V^*$…
user124910
  • 3,007
21
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2 answers

How to find the basis for a vector space?

I've been given the following as a homework problem: Find a basis for the following subspace of $F^5$: $$W = \{(a, b, c, d, e) \in F^5 \mid a - c - d = 0\}$$ At the moment, I've been just guessing at potential solutions. There must be a better…
Casey Patton
  • 1,453
20
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2 answers

vector spaces whose algebra of endomorphisms is generated by its idempotents

Let $V$ be a $K$-vector space whose algebra of endomorphisms is generated (as a $K$-algebra) by its idempotents. Is $V$ necessarily finite dimensional? EDIT (Jul 26 '14) A closely related question: Is there a field $K$ and a $K$-vector space whose…
Pierre-Yves Gaillard
  • 19,712
20
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2 answers

Integral around unit sphere of inner product

For arbitrary $n\times n$ matrices M, I am trying to solve the integral $$\int_{\|v\| = 1} v^T M v.$$ Solving this integral in a few low dimensions (by passing to spherical coordinates) suggests the answer in general to be…
user7530
  • 49,280
20
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4 answers

Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional.

Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional. Clearly it's infinite dimensional, because if you consider say $P (\mathbb{F})$ on $[0,1]$, then there are an…
St Vincent
  • 3,070
20
votes
1 answer

Direct Sum of vector subspaces

How is direct sum of two vector subspaces different from the sum of two vector subspaces i.e. how is $X\oplus Y$ different from $X + Y$, where $X, Y$ are subspaces.
Hashtag
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20
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5 answers

how to extend a basis

This is a very elementary question but I can't find the answer in my book at the moment. If I have, for example, two vectors $v_1$ and $v_2$ in $\mathbb R^5$ and knowing that they are linear independent , how can I extend this two vectors to a basis…
sigmatau
  • 2,622
20
votes
2 answers

Prove that T is diagonalizable if and only if the minimal polynomial of T has no repeated roots.

Prove that T is diagonalizable if and only if the minimal polynomial of T has no repeated roots. EDIT: ( Over $\Bbb C $ ) though it is obvious i am working over $\Bbb C $ as one of my statements is not true over $ \Bbb R $ I would like a better…
Faust
  • 5,669
20
votes
1 answer

Proof that Eigenvalues are the Diagonal Entries of the Upper-Triangular Matrix in Axler

This is 5.18 from Axler's Linear Algebra Done Right: Theorem: Suppose $T \in L(V)$ has an upper-triangular matrix with respect to some basis of $V$. Then the eigenvalues of $T$ consist precisely of the entries on the diagonal of that…
mathnoob
  • 1,319
20
votes
3 answers

Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.

A function $T:V \rightarrow W$ is additive if $T(x+y) = T(x) + T(y)$ for every $x, y \in V$. Prove that there exists an additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear. My attempt: Let $T$ be the function $T: \mathbb{R}$…
elbarto
  • 3,356
20
votes
1 answer

How can LU factorization be used in non-square matrix?

In my textbook, there is some information about LU factorization of square matrix $A$, but not about non-square matrix. How can LU factorization be used to factorize non-square matrix?
Susan Pioloco
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