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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 $Dim(W) \leq k$

Let be $V$ a vector space over a field $F$ with finit dimension. Let be $T$ a lineal operator in $V$. Supose that the characteristic polynomial of $T$, $p(x)$, is of the form $p(x)=(x-c)^{k}g(x)$ with $k > \in \mathbb{N}^{+}$, $c \in F$ and $g(c)…
luisegf
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Orthogonal and Orthonormal Matrix

I know that the columns of an orthogonal matrix are perpendicular to each other and additionally if the columns have unit length then they are orthonormal. But my professor states that the columns of an orthogonal matrix form an orthonormal basis?…
Orpheus
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Dot product properties

I want to prove or contradict the following claim: If we take two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ in $\mathbb{R}^{d}$ ($d$ isn't neccesarily 2, so geometric proofs aren't available) and the angle between them, which is defined by…
Roei Sarussi
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Linear function preserving the Gram determinant

In Euclidean space $X$ the Gram's determinant of a system of vectors $x_1,...,x_k\in X$ is called the determinant of $k\times k$ matrix $ [\langle x_i,x_j \rangle]$: $ G(x_1,..,x_k)=\det[\langle x_i,x_j \rangle]. $ In $n$ dimensional Euclidean…
Richard
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Showing no non-trivial t-invariant subspace has a t-invariant complement.

The question is from Hoffman and Kunze Let $T$ be a linear operator on a finite-dimensional vector space $V$. Suppose that: (a) the minimal polynomial for $T$ is a power of an irreducible polynomial ; (b) the minimal polynomial is equal to the…
AvatarOfChronos
  • 1,701
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$f\colon \Bbb R^3 \to \Bbb R^3 $ be defined by $f(x_1,x_2,x_3)=....$

I am stuck on the following problem: Let $f\colon \Bbb R^3 \to \Bbb R^3 $ be defined by $f(x_1,x_2,x_3)=(x_2+x_3,x_3+x_1,x_1+x_2).$ Then the first derivative of $f$ is : 1.not invertible anywhere 2.invertible only at the origin 3.invertible…
user52976
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Can $y=10^{-x}$ be converted into an equivalent $y=\mathrm{e}^{-kx}$?

I was dealing with the values: | Digits | Expression | Value | |--------|------------|-----------------------| | 1 | 10⁻¹ | 0.1 | | 2 | 10⁻² | 0.01 | | 3 | 10⁻³ |…
Ian Boyd
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Eigenvalues of a Hermition Matrix do not cross

Wikipedia's article on avoided crossing asserts that "The eigenvalues of a Hermitian matrix depending on N continuous real parameters cannot cross except at a manifold of N-2 dimensions." If it's true, does anyone have an elegant proof of this…
ChickenGod
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A question on proving an equation to be an $n$-linear system in linear algebra

While studying Determinants from text book Hoffman and Kunze, I have a in an argument in a theorem whose reasoning is not provided . Questions: 1st question is in underlined part of theorem. It's image : How did authors deduced that $A_{ij}…
user775699
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Is it true that there is an interesting entry in each row of a matrix with nonzero determinant?

We call an entry of an $ n × n $ matrix with nonzero determinant interesting, if by changing this entry (and only this) the determinant of the matrix can be made $0$. Is it true that there is an interesting entry in each row of a matrix with nonzero…
Subu
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Relating "change of coordinates" to change of basis - how to find change in representations of vectors

I've been studying about change of basis in $\mathbb{R}^2$ (could be $\mathbb{R}^n$ but sticking to $\mathbb{R}^2$ for simplicity) - how it affects representations of vectors, metrics and endomorphisms. Let's say I start with a basis…
Shirish Kulhari
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Sums of a set of symmetric matrices

Say we have a set of symmetric $n \times n$ matrices $M_i$ for $1 \leq i \leq k$, elements in $\mathbb{R}$. Suppose that for every $\boldsymbol{\lambda} = (\lambda_1, \dots , \lambda_k) \in \mathbb{R}^k$ we have that the kernel of…
I Like Trams
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Primary Decomposition Theorem: What are the characteristic polynomials of the maps of the decomposition?

Let $T$ be an operator on a finite dimensional vector space $V$. Suppose that the characteristic polynomial of $T$ is $$\chi(t)=f_1^{n_1}(t)\cdots f_k^{n_k}(t)$$ where $f_1,\ldots,f_k$ are distinct irreducible polynomials, and suppose the minimal…
Spenser
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Prove T is normal if and only if T = T1 + iT2, where T1 and T2 are selfadjoint operators which commute.

Got this question for homework, im having troubles to prove one side of it The question: Prove T is normal if and only if T = T1 + iT2, where T1 and T2 are selfadjoint operators which commute. $<=$ lets assume we have T1, and T2 such as…
dave
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Positive and unitary operator over a finite dimensional vector space

I have got the below question as part of an homework assignment: Let $T$ be a linear operator on the finite-dimensional inner product space $V$, and suppose $T$ is both positive and unitary. Prove $T = I$. I tried to prove it the following…
dave
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