Mathematics Stack Exchange
Mathematics Stack Exchange

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
11
votes
1 answer

How do I get the parametric form solution of a linear system from reduced row-echelon form?

I have the following system of equations: x1 + 6x2 + 2x3 - 5x4 = 0 -x1 - 6x2 - x3 - 3x4 = 0 2x1 + 612x2 + 5x3 - 18x4 = 0 and I understand that it translated into the following matrix: 1 6 2 -5 0 -1 -6 -1 -3 0 2 …
mimicocotopus
  • 499
11
votes
3 answers

Proving that $\mathbf{W}$+$\mathbf{W^{\perp}}$=$\mathbb{R^{n}}$

I am trying to prove that given a subspace $\mathbf{W}$ in $\mathbb{R^{n}}$, the subspace and its orthogonal complement 'cover' whole of $\mathbb{R^{n}}$ through '+' where we define $\mathbf{W}$+$\mathbf{W^{\perp}}$ as linear combinations of vectors…
LinAlgStudent
  • 2,039
11
votes
3 answers

Basic Linear Algebra Proof - Orthogonal Vectors

Prove that if $\mathbf{u}$ and $\mathbf{v}$ are nonzero orthogonal vectors in $\Bbb R^n$ they are linearly Independent. I've struggled with this a bit, here is what I know so far: Suppose $\mathbf{u}$ and $\mathbf{v}$ are orthogonal. Then…
Ben Anderson
  • 681
11
votes
1 answer

To Interpret Solving Systems of Linear Equations Geometrically in Terms of Linear Algebra

I never really understood basic Gaussian elimination & solving systems of equations once I learned some actual linear algebra. I thought this was due to me missing some fundamental aspect of the subject that some book would eventually illuminate for…
sponsoredwalk
  • 1,271
11
votes
2 answers

Why the spectral theorem is named "spectral theorem"?

"If $V$ is a complex inner product space and $T\in \mathcal{L}(V)$. Then $V$ has an orthonormal basis Consisting of eigenvectors of T if and only if $T$ is normal".   I know that the set of orthonormal vectors is called the "spectrum" and I guess…
Hiperion
  • 1,763
11
votes
1 answer

Finding null space of matrix.

I need to make sure I'm understanding this correctly. I skipped a few steps to reduce typing, but let me know if I need to clarify something. Question asks: Find $N(A)$ for $A$ = \begin{bmatrix} -3 & 6 & -1 & 1 & -7 \\ 1 & -2 & 2 &…
cisco
  • 339
11
votes
6 answers

Find a $4\times 4$ matrix $A$ where $A\neq I$ and $A^2 \neq I$, but $A^3 = I$.

Give an example of a $4 \times 4$ matrix where $A \neq I$, $A^2 \neq I$, and $A^3 = I$. I found a $2 \times 2$ matrix where $A \neq I$ and $A^2 = I$, but this problem is more complex and has me completely stumped.
Grace C
  • 335
11
votes
2 answers

Number of matrices on $\mathbb Z_{p}$ with a given characteristic polynomial

How can I find the number of n×n matrices on $\mathbb Z_{p}$ with a given characteristic polynomial? for example: If $p$ is a prime number s.t. $p \equiv 3 \pmod 4$, then number of $2\times 2$ matrices on $\mathbb Z_{p}$ that their characteristic…
Mojee KD
  • 420
11
votes
5 answers

Expanding a basis of a subspace to a basis for the vector space

I'm not really sure how to extend a basis. I'm trying to do the following question. Consider the subspace $ W = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = -x_4, x_2 = x_3\}$ of $ \mathbb{R}^4$. Extend the basis $\{(0,2,2,0),(1,0,0,-1)\}$ of $W$…
user1520427
  • 625
11
votes
2 answers

The image of the transpose of $A^\text{T}$ is the orthogonal complement of its kernel

Suppose $V$ is a finite dimensional vector space over $\mathbb{K}$. Let $A$ be a linear map. I am trying to prove that $$\operatorname{Im}A^\text{T}=(\operatorname{Ker}A)^{\perp}$$ I know one direction: $\operatorname{Im}A^\text{T} \subset…
user335468
  • 639
11
votes
1 answer

If $A$ has eigenvalues $\lambda_1,...,\lambda_n$, is there a relationship between the eigenvalues of $A$ and $\hat{A}$

Suppose a square matrix $A$ has eigenvalues $\lambda_1,...,\lambda_n$. Note that A is a $n\times n$ matrix where $n$ is even. Let $\widehat{A}$ be a matrix that is obtained from $A$ only by multiplying every second row in $A$ by $-1$. Is there a…
eggSand
  • 111
11
votes
1 answer

Action of a matrix on the exterior algebra

I read in a paper that if $M$ is a real square matrix of size $n$, then we can consider the action of $M$ in the third exterior algebra $\Lambda^3 \mathbb{R}^n$, and the matrix of this action are the $3\times 3$ minors of $M$. Here I am not clear…
mapping
  • 685
11
votes
3 answers

Trace of AB = Trace of BA

We can define trace if $A =\sum_{i} \langle e_i, Ae_i\rangle$ where $e_i$'s are standard column vectors, and $\langle x, y\rangle =x^t y$ for suitable column vectors $x, y$. With this set up, I want to prove trace of AB and BA are same, so it's…
Myshkin
  • 35,974
  • 27
  • 154
  • 332
11
votes
3 answers

The dimension of centralizer of a Matrix.

Let $A$ be a $n\times n$ matrix with characteristic polynomial $$(x-c_{1})^{{d}_{1}}(x-c_{2})^{{d}_{2}}...(x-c_{k})^{{d}_{k}}$$ where $c_{1},c_{2},...,c_{k}$ are distinct. Let $V$ be the space of $n\times n$ matrics $B$ such that $AB=BA$. How to…
neelkanth
  • 6,048
  • 2
  • 30
  • 71
11
votes
1 answer

invertible $\{0, 1\}$ matrix

For an $n\times n$ matrix $A$ with entries from $\{0,1\}$. What is the maximum number of 1's such that $A$ is invertible? If $n=2$, the answer is $3$. If $n=3$, the answer is $7$. Is there a formula for general $n$?
Sunni
  • 4,536
1 2 3
…
99 100