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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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Is there a useful matrix identity for this?

Suppose we have: $$ S = \left[ \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{matrix} \right]$$ $$ X_0 = \left[ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ j & k &…
user847663
  • 175
4
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2 answers

For each given subspace W there is one and only one row-reduced echelon matrix that has W as its row space.

Let m and n be positive integers and let F be a field. Suppose W is a subspace of $ {F ^{n}}$ and $ dim W \le m$. Prove that there is precisely one m x n row-reduced echelon matrix over F which has W as its row space.
darkgbm
  • 1,810
4
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2 answers

Let $ABC$ be a well-defined product of matrices. Suppose that $A,C$ are both invertible. Prove that $rank(ABC)=rank(B)$.

Not sure where to go with this... I've done the following, though: $A$ and $C$ must be square matrices if they are invertible. Let $A$ be an $m\times m$ matrix, and let $C$ be an $n\times n$ matrix. Let $B$ be a $m\times n$ matrix so that $ABC$ is…
Mirrana
  • 9,009
4
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4 answers

Finding the characteristic polynomial of a linear transformation

Let $T: M_n(\Bbb{R}) \to M_n(\Bbb{R})$ be a linear transformation defined by $T(A)=A^t+A$. What is the characteristic polynomial of $T$? If I use the basis $E_{ij}$, I get $T(E_{ij})=E_{ij}+E_{ji}$, but don't know how to compute the characteristic…
Gobi
  • 7,458
4
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2 answers

How prove this $(x,y)H\binom{x}{y}\ge p(x^2+y^2)$

let $H_{2\times 2}$ is positive definite matrix, show that: There exist $p>0$, such $$(x,y)H\binom{x}{y}\ge p(x^2+y^2)$$ My try: let $$H=\begin{bmatrix} a_{1}&a_{2}\\ b_{1}&b_{2} \end{bmatrix}$$ and such…
math110
  • 93,304
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Dimension of a space of matrices

Let $m,n\in\mathbb{Z}$ and $r<\min$(m,n). Denote by $M$ the set of $m\times n $ matrices over a field $k$, and let $M_r$ be the subset of matrices of rank at least equal to $r$. Now fix a matrix $A\in M_r$ and a subspace $W\in G(n-r,n)$ of dimension…
Abramo
  • 6,917
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Prove that a linear transformation is invertible if and only if its associated matrix is invertible.

Let $V$ be a finite dimensional vector space, $\beta$ an ordered basis of $V$, $T$ a linear operator on $V$ and $A$ the associated matrix of $T$ in the basis $\beta$. I have to prove that $T$ is invertible if and only if $A$ is invertible. I was…
Twnk
  • 2,436
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3 answers

Proving that this set is linearly independent

Ler $V=P(\Bbb R)$ be the vector space of the polynomials with real coefficients, on the field of real numbers $\Bbb R$. For $i \geq 1$, let $T_i(f)=f^{(i)}$ the $i$th derivate of $f$. I have to show that for any $n \in \Bbb N$, $\{T_1, T_2,...,…
Twnk
  • 2,436
4
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3 answers

Is $V$ a vector space under these operations?

Let $$V=\{x \in \mathbb{R} : x>0\}$$ For $x,y,a \in \Bbb{R}$, define $x\oplus y=xy$ and $x\odot a=x^a$. Is $V$ a vector space under these operations? Justify your answer.
v sadegh
  • 59
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2 answers

prove that $T$-cyclic subspace of $V$ generated by $x$ is $T$-invariant

Let $T$ be a linear operator on a vector space $V$, and let $x$ be a non-zero vector in $V$. The subspace, $$W = \operatorname{span}(\{x,T(x),T^2(x),\ldots\})$$ I have to prove that $W$ is a $T$-invariant subspace of $V$ and also that it is the…
johny
  • 1,609
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2 answers

orthogonal basis of eigenvector

Let $A:\mathbb R^n \to \mathbb R^n $ be a diagonalizable matrix. From the definition, this means that there exist a basis of eigenvectors. Using the Gram.Schmidt Algorithm, one can prove that for every subspace of $\mathbb R^n$ there exist an…
Shanks
  • 789
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5 answers

How do you prove that vectors are linearly independent in $ \mathcal{C}[0,1]$?

I'm presented with the question: Show that the given vectors are linearly independent in $\mathcal{C}[0,1]$: $x^{3/2}, x^{5/2}$ I'm having a terrible time understanding linearly algebra in general. I think my part of my problem with this question…
rphello101
  • 345
4
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2 answers

Rowspan of 0-1 Matrix

Let $A$ be an $n \times n$ matrix with entries 0 or 1 with the following properties: Every column has a nonzero entry Every row has a nonzero entry No rows are repeated Is it true that the vector $(1, \ldots, 1)$ lies in the span of the rows of…
Hans Parshall
  • 6,246
4
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1 answer

How prove this the vector is an eigenvector with eigenvalue $1$

Let $A$ be an element of $SO_{3}$,show that if it is defined,the vector $$((a_{23}+a_{32})^{-1},(a_{13}+a_{31})^{-1},(a_{12}+a_{21})^{-1})^T$$ is an eigenvector with eigenvalue $1$ I found this Closed-form for eigenvectors of rotation matrix I kown…
math110
  • 93,304
4
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4 answers

Matrix Algebra Question (Linear Algebra)

Find all values of $a$ such that $A^3 = 2A$, where $$A = \begin{bmatrix} -2 & 2 \\ -1 & a \end{bmatrix}.$$ The matrix I got for $A^3$ at the end didn't match up, but I probably made a multiplication mistake somewhere.
inabajunes
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