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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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Whats wrong with this proof of Cayley-Hamilton

There are a handful of longish proofs for Cayley-Hamilton, but I haven't seen one that goes along the following lines. What's wrong in my thinking. Take a real square matrix $A$, with Jordan decomposition $D=Q^{-1}AQ$. Call the eigenvalues of $A$…
user1816847
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Linear span of functions

Maybe this is something basic but I am not familiar with the term "linear span" and in one question it mentions the linear span of the functions $f_n(t) = e^{nt}$, $n = 0,1,2,...$ $t \in [a,b]$ What I understood is linear span is the set of linear…
marvinthemartian
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If in an inner product space $\langle x,u\rangle=\langle x,v\rangle$ for all $x$, show that $u=v$.

If in an inner product space $\langle x,u\rangle=\langle x,v\rangle$ for all $x$, show that $u=v$. This seems obvious to me, so how do I prove it? Proof by contradiction maybe? Any suggestions would be nice.
Desperate Fluffy
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If the dot product between two vectors is $0$, are the two linearly independent?

If we have vectors $V$ and $W$ in $\mathbb{R^n}$ and their dot product is $0$, are the two vectors linearly independent? I can expand $V_1 \cdot V_2 = 0 \Rightarrow v_1w_1+...+v_nw_n = 0$, but I don't understand how this relates to linear…
Joel B
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How to transfrom my equation to $Y=KX^2$

In general , $$\vec{C}(u)=\vec{a_0}+\vec{a_1} u+\vec{a_2} u^2$$ is a parabolic arc between the points $\vec{a_0}$ and $\vec{a_0} + \vec{a_1} + \vec{a_2}$. So I'd like to prove it by myself: My trial as below: $\vec{a_i}=(x_i,y_i)^T$ $\Rightarrow$…
xyz
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Which one is true? (CSIR)

Let $a,b,c$ be a positive real number such that $b^2+c^2
S. Pitchai Murugan
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Don't understand what this question about solutions of linear equations is asking.

If there exists a solution for the linear equation $A x = \left( \begin{array}{lll} 1 \\ 1 \\ 1 \end{array} \right)$, then there also exists a solution for $Ax = \left( \begin{array}{lll} 1 \\ 2 \\ 1 \end{array} \right)$. I have a list of…
nofe
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How to tell if a columns of matrix are linear dependent?

How can it be seen if the following matrix is linear dependent? Let $A= \begin{bmatrix} 0 & -3 & 9& \\ 2&1& 7 \\ -1& 4 &-5 \\ 1&-4&-2 \end{bmatrix} $ First operation I perfomed was switch r1 to r4 and -2r1+r2 $A= \begin{bmatrix} 1 & -4 & -2& …
Fernando Martinez
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Any easier and direct method to prove $\|P\|_2=1$ and $\|Px\|\le\|x\|$ for orthogonal projector $P$

I know complicated ways to prove them. But wonder if there is any easier and direct method to prove $\|P\|_2=1$ and $\|Px\|\le\|x\|$ for orthogonal projector $P$ just given the fact $P^T=P$ and $P^2=P$. Please help. Thank you.
Qiang Li
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Find whether vector w belongs in the span

$$v_1=[1,0,1,2]$$ $$v_2 = [0,1,1,3]$$ $$v_3 = [2,1,3,7]$$ $$w = [1,2,3,4]$$ We are supposed to determine if $w$ is in $\operatorname{span}(v_1,v_2,v_3)$.
bob
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Normal and Lower triangular matrix implies diagonal matrix

A lower triangular complex matrix $A$ satisfies $AA^*=A^*A$. I would like to show that $A$ is diagonal. I know there exists a unitary matrix $P$ such that $PAP^*$ is diagonal. But I don't know how to show $A$ itself is diagonal.
user9352
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Dimension of vector space of 2x2 skew symmetric matrices

I had a question about the dimension of this subspace. This was related to a problem that had a case of $n\times n$ matrices, but I accidentally read it as the special case of $2\times 2$, but never the less the answer to this question should help…
user1236
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Solving system of linear equations in the field $Z_p.$

Consider $5x+3y=4$ and $3x+6y=1.$ List the set of primes for which this system of linear equations does not have a solution in the field $Z_p.$
Mirunalini_UML
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Prove that $\sin(x+a), \sin(x+b),\sin(x+c), \hspace{5pt} a,b,c \in \mathbb{R}$ are linearly dependent

I just want an answer verification (or not). We have that: $\begin{array}[t]{l} \sin(x+a)=\cos a\cdot \sin x + \sin a \cdot \cos x\\ \sin(x+b)=\cos b\cdot \sin x + \sin b \cdot \cos x\\ \sin(x+c)=\cos c\cdot \sin x + \sin c \cdot \cos…
thanasissdr
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Does $\forall x\ne 0: x^TAx>0$ means all eigenvalues of $A$ are real?

Let $A\in\mathbb{R}^{n\times n}$ Does $\forall x\ne 0,x\in \mathbb{R}^n: x^TAx>0$ means $A$ has only real eigenvalues (roots of the characteristic polynomial are all real)?
anonymous67
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