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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
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Linear maps (complex)

$(1)$ Let $a, b\in \mathbb C$ and $\alpha: \mathbb C^2 \to \mathbb C$ be given by $(x, y)\mapsto ax + by.$ $\quad(a)\quad$ Show that $\alpha$ is a $\mathbb C$- linear map. What condition(s) you have to check? $\quad(b)\quad$ For what values of $a$…
mrHarry
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Transpose for an infinite dimensional vector space

Suppose that $V$ and $W$ are 2 finite dimensional vectors spaces and $T$ is a linear transformation such that $T : V \rightarrow W$. Then $T \rightarrow T^t$ can be seen as an isomorphism of $L(V,W)$ into $L(W^\ast,V^\ast)$. Now suppose that $V$ and…
darkgbm
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Minimal polynomial of an inverse given the min polynomial of the matrix

Question: Given that $A$ is an invertible matrix and $m_A(x)=a_0+a_1x+...+a_nx^n$ find $m_{A^{-1}}$. Thought: If I put A in the polynomial then it's equal to 0, then I multiply the entire equation by (A^-1)^n , then I get some kind of polynomial:…
jreing
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Showing the linear independence between two row equivalent matrices

I can prove that if A and B are row equivalent matrices, then the column vectors of A are linearly independent iff the column vectors of B are linearly independent. However, does this result also hold for row vectors? That is, is it true that if A…
Trts
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For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , show $V= \ker T^n \oplus $ Im $T^n$

I'm looking for the shortest and the clearest proof for this following theorem: For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , I need to show $V= \ker T^n \oplus $ Im $T^n$. Any hints? I don't know…
user6163
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Let $A$ be an orthogonal $n\times n$ matrix. Show that $\|A\vec x\|=\|A^{-1}\vec x\|$ for any vector $x$ in $\mathbb R^2$

Let $A$ be an orthogonal $n\times n$ matrix. Show that $\|A\vec x\|=\|A^{-1}\vec x\|$ for any vector $\vec x$ in $\mathbb R^2$ I want to show that $\|A\vec x\|=\|A^{-1}\vec x\|=\|\vec x\|$ I tried to show that since $A^TA=I$, then using…
user95087
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Is a $1\times 1 $ matrix a scalar?

Intuitively I used to think that a $1\times 1$ matrix is simply a scalar number, I also saw this statement in books. However when I think about it now it doesn't make sense to me because of one problem. Let $I$ be the $2\times 2 $ identity matrix,…
Slugger
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Irreducible characteristic polynomial of a linear transformation

I am trying to understand why the linear transformation $T$ corresponding to the companion matrix of the minimal polynomial of an irreducible polynomial over $\mathbb{Q}[x]$ (for instance $x^3-x-1$) has no non-trivial $T$-invariant subspaces. I know…
user7980
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Prove two spans are equal

How would I go about showing that two spans are equal? I was thinking that I could somehow prove that each is a subset of the other but I'm not sure how. If sp(A) is a subset of sp(B), I could prove that sp(B) is a subset of sp(A). Could I…
codedude
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Determine homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $y = 2x - 6$

Determine the homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $ y = 2x - 6$. I use $mx - y +b =0$: $\text{slope} = m$, $\tan(\theta)= m$ intersection with the axes: $x =0$ is $y = -b$ and $y…
belo gadelo
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Bijection between column space and row space

Suppose that $A_{mn}$ is a matrix over some field, and that $C, R$ is its column space and row space, without using the fact that $rank(C) = rank(R)$, can we show that, there exists a bijection between $C$ and $R$?
Not an ID
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When is $1\vec{u} \neq \vec{u}$?

In a linear algebra textbook they define a vector space to be a nonempty set $V$ of objects that satisfy certain properties. One of these properties is that $\forall\vec{u}\in V(1\vec{u}=\vec{u})$ The only way I can think of that property NOT…
scott_fakename
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A linear operator $T: V \rightarrow V$ commuting with all linear operators is a scalar multiple of the identity.

Let $\mathbb{K}$ a field, $V$ a vector space over $\mathbb{K}$. If $T:V\to V$ commutes with all other linear operators $V \to V$, then there exists $\lambda \in \mathbb{K}$ such that $T= \lambda I$, where $I$ is the identity V.
Croos
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prove that $T^2=T$ diagonalizeable without using Jordan

Question: Given that $T:V \to V$ and $T^2=T$ prove that $T$ is diagonalizable. What I know: $T^2-T=0=T(T-I)$. $\operatorname{Im}(T-I) \subseteq\operatorname{Ker}(T)$ therefore $\dim V=\dim\operatorname{Ker}(T-I)+\dim\operatorname{Im}(T-I)\leqslant…
jreing
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3D Linear equation problem

I have two points in 3D space: point $A = (1,2,3)$ point $B = (4,7,6)$ I want to find a third point between the two, where $z = 5$ So, point $C = (x,y,5)$ How can I calculate $x$ and $y$ for point $C$? Thanks.
Mustafa
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