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
7
votes
1 answer

Geometric interpretation of Gauss elimination

Solution of a set of linear equations is finding a point of intersection of all planes represented by those equations. But how can we relate it to gauss elimination method ? Suppose we have equations like- x + 3y = 4 (A) 2x - 6y = 8 (B) We are…
Krrish Raj
  • 185
7
votes
4 answers

Number of matrices whose square is identity

How many matrices are such that $A^2 =I$, where $A$ is a $2\times2$ matrix and $I$ is a $2\times2$ identity matrix? I can only think of the identity and it negative are they more? Is it an application of Cayley-Hamilton theorem. I have seen a…
Vaolter
  • 1,711
7
votes
2 answers

Prove that $(A+B)^n = 0$

Let $A$ and $B$ be $n \times n$ real matrices such that $A^n = B^n = 0$ and $AB = BA$. Prove that $(A+B)^n = 0$. We have $$(A+B)^n = A^n+A^{n-1}B\binom{n}{1}+\cdots+AB^{n-1}\binom{n}{n-1}+B^n.$$ I tried proving this for just the case $n = 2$. We…
user19405892
  • 15,592
7
votes
1 answer

Why is this not an isomorphism?

Let $T(f(t))$= $\begin{bmatrix} f(0) & f(1)\\ f(2) & f(3) \end{bmatrix}$ from $P_2$ to $\mathbb{R}^{2\times 2}$. To show that it is not an isomorphism, I need to show that either kernel of the transformation is not equal to the zero element…
user7008
7
votes
1 answer

Inner product for vector space over arbitrary field

The definition of an inner product in Linear Algebra Done Right by Sheldon Axler assumes that the vector space is over either the real or complex field. PlanetMath makes the same assumption. Is there a definition of an inner product over, for…
Snowball
  • 3,078
  • 18
  • 37
7
votes
1 answer

Number of non singular matrices over a finite field of order 2

I have to find out the number of $3×3$ non singular matrices over a field of order $2$. I tried in the following way. First to find out a non singular matrix $A,$ clearly any row of $A$ can't be full of $0$s. So the first row (say) can be filled up…
hiren_garai
  • 1,193
7
votes
3 answers

How do you find a basis for the set of all $3 \times 3$ matrices whose rows and columns add up to zero?

If $W$ is the set of all $3 \times 3$ matrices whose rows and columns add up to zero, how would you find a basis for this? There seem to be so many scenarios we'd need to cover, I can't find a way to succinctly find the answer / represent it…
dmonopoly
  • 391
7
votes
1 answer

Linear transformations and norm

I am studying Linear Algebra II, and I came across several questions in which, for a certain linear transformation ($T\colon\mathbf{V}\to\mathbf{V}$) I was told that: $$||T(a)|| \leq ||a||.$$ I am not completely certain how to use this information.…
Hila
  • 1,919
7
votes
0 answers

Intuition behind the change of basis matrix

My linear algebra class isn't particularly rigorous and my professor doesn't really provide much intuition for most of the theorems we learn, either. Because of this, I've made an effort to make sense of the theorems beyond "the math works out to…
youngsmasher
  • 457
  • 4
  • 9
7
votes
2 answers

Are there other Identity Matrices?

Is there only one identity matrix $$\begin{pmatrix} 1&0&...&...&0\\0&1&0&...&0\\...&0&1&...&0\\...&...&0&1&0\\...&...&...&0&1\end{pmatrix}$$ etc.. Or are there different identity matrices for other bases? A textbook example asks if $[T]_{\beta} =…
CodyBugstein
  • 1,642
7
votes
4 answers

Disjoint Union of Subsets and Direct Sum of Subspaces (Clarify Explanation)

Can someone clarify for me the side note found in Axler's Linear Algebra Done Right, that is: Direct sums of subspaces are analogous to disjoint union of subsets. I am not sure what exactly is a disjoint union of subsets (having checked online…
mathnoob
  • 1,319
7
votes
6 answers

Show that the vector space of all continuous real-valued functions is infinite-dimensional

Show that the vector space $C(\Bbb R)$ of all continuous functions defined on the real line is infinite-dimensional. I get that if $C(\Bbb R)$ contains an infinite-dimensional subspace, then it is infinite-dimensional, but how do I prove that?…
Lola1984
  • 313
7
votes
3 answers

Prove that if Ax = b has a solution for every b, then A is invertible

I am interested in the case that $A$ is a matrix over a commutative ring, not necessarily a field. Is it still true that if $Ax = b$ has a solution for every $b$, then $A$ is invertible? I know that in the general setting, $A$ having the trivial…
fdzsfhaS
  • 1,105
  • 10
  • 29
7
votes
2 answers

Check if polynomials are linearly independent.

I would like to check if polynomials $1, 1+t^2, 1+t+t^2$ are linearly independent. My idea is: $1 \to [1,0,0]$ $1+t^2\to [1,1,0]$ $1+t^2+t^3 \to [1,1,1]$ And now $\left( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{array} \right)$ I…
Happy man
  • 395
7
votes
2 answers

Multiply $1\times 3$ matrix by corresponding numbers

For example, I want to this to happen: $$\begin{bmatrix}1& 2& 3\end{bmatrix}\times\begin{bmatrix}2& 3& 4\end{bmatrix} = \begin{bmatrix}2& 6& 12\end{bmatrix}$$ It's not exactly matrix multiplication, but I hope you can see what I'm getting at. Is…
Raven
  • 173
1 2 3
…
99 100