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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 Trace and the main diagonal of a matrix are distinguished

Let $A$ be a square $n \times n$ over a field (say $\mathbb{R}$ or $\mathbb{C}$). As we know, the main diagonal $(a_{1,1},...,a_{n,n})$ is important in linear algebra while the off-diagonal is far less important. The deep question is why? We try to…
LinAlgMan
  • 2,924
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Decompose a real symmetric matrix

Prove that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $A$ as its diagonal elements. I can see that all…
Not an ID
  • 877
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How prove $A^2=0$,if $AB-BA=A$

let $A_{2\times 2}$ matrix, and The matrix $B$ is order square,such $$AB-BA=A$$ show that $$A^2=0$$ My idea: since $$Tr(AB)=Tr(BA)$$ so $$Tr(A)=Tr(AB-BA)=Tr(AB)-Tr(BA)=0$$ Question:2 if $A_{n\times n}$ matrix,and the matrix $B$ is order…
math110
  • 93,304
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Linear map from zero vector to zero vector.

I am reading an introduction on linear maps in my text book on linear algebra. The following statements are made: Suppose $G_1 (\vec{u}) = (x_1 + 2x_2 + 3x_3 + 1, 4x_1, 9x_3)$ Then we can use the following property of linear maps. Let $\lambda = 0$…
Lukas Arvidsson
  • 2,376
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Why not use the identity matrix instead of the Kronecker delta?

The Kronecker delta is defined as : $$\delta_{mn} = \begin{cases} 1 & \text{if }m=n,\\ 0 & \text{if }m\neq n. \end{cases}$$ This is equal to the matrix $E_n$ which is a matrix with the diagonal filled with ones. Why not use $E_n$ instead of…
VVV
  • 2,695
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Prove that if $ u \cdot v = u \cdot w $ then $v = w$

I've tried putting it up as: $$ [u_1 v_1 + \ldots + u_n v_n] = [u_1 w_1 + \ldots + u_n w_n] $$ But this doesn't make it immediately clear...I can't simply divide by $u_1 + \ldots + u_n$ as these ($u$, $v$ and $w$) are vectors... Any hints?
user3200098
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Exponential of matrix with negative entries only on the diagonal

Suppose I have a matrix $A$ with real entries such that the off-diagonal entries of $A$ are positive or zero. (The diagonal entries may be positive, negative or zero.) From doing a few examples in Python, it looks like the following might be true of…
N. Virgo
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When is every matrix in the span of two matrices singular?

Given two square matrices $A, B$, when is $$\det(A+tB) = 0$$ for all $t\in \mathbb{R}$? An easy sufficient condition is that $A$ and $B$'s kernels have nontrivial intersection. Per Henning's comment below, this is not also necessary. Does there…
user7530
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How do I find a basis for the subspace of $\mathbb{R}^4$ spanned by the four vectors below:

$v_1=(1, 1, 2, 4), v_2=(2, -1, -5, 2),v_3 = (1, -1, -4, 0), v_4=(2, 1, 1, 6).$ I started with writing down the matrix and through row reduction ended with the solution $x_1=4/3 + (1/3)x_3$ and $x_2 = 1/3 - (2/3)x_3$, $x_3$ is free. I think the basis…
Noor Aslam
  • 255
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Construct an endomorphism $f$ such that $f\circ f=-Id$

Let $E$ be a real vector space of finite dimension $n$ and $f$ an endomorphism such that $$f\circ f=-Id_E$$ Show that $n = \dim (E)$ is an even integer Assume $n$ is even, $n=2p$. Construct an endomorphism $f$ such that $f\circ f=-Id$ 1) I…
user117932
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Showing that $\{ 1, \cos t, \cos^2 t, \dots, \cos^6 t \}$ is a linearly independent set

The problem is: Show that $\{ 1, \cos t, \cos^2 t, \dots, \cos^6 t \}$ is a linearly independent set of functions defined on $\mathbb{R}$. The problem expects the student to use a computer program such as Matlab. To solve the problem I created a…
Jodles
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Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set

Problem 4.3.32 in Linear Algebra, Lay: Let $V$ and $W$ be vector spaces, let $T:V\to W$ be a linear transformation, and let $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ be a subset of $V$. Suppose that $T$ is a one-to-one transformation [...]. Show…
Jodles
  • 567
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Proving the product of two non singular matrices is also non singular.

I am having trouble with a proof for linear algebra. Could somebody explain to me how to prove that if $A$ and $B$ are both $n\times n$ non singular matrices, that their product $AB$ is also non singular. A place to start would be helpful. Thank…
cogle
  • 173
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Two proofs about invertible Matrix and row equivalent to the identity matrix.

2 proofs: 1. A matrix n*n is invertible if and only if it's row equivalent to the identity matrix. 2. A matrix n*n is not invertible if and only if it is row equivalent to a matrix with zero row. (proofs without determinant of course) I know…
rab2004
  • 331
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How to prove that there exists a $B$ such that $A=B^2$

If matrix $A$ is such that $A+A^T$ is a positive definite matrix, show that there exists a $B$ such that $A=B^2$, where $B+B^T$ is a positive definite matrix. My try: since $A+A^T$ is positive matrix, then exists $Q$…
math110
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