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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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proving:Lamda is an eigenvalue of T if and only if $T-\lambda \mathrm{I}$ is not injective.

I'm trying to prove a theorem from "linear Algebra Done right" by Sheldon Axler. $\lambda$ is an eigenvalue of T if and only if $T-\lambda \mathrm{I}$ is not injective. I'm a bit confused about what we assume to be true. So, I'm not sure what to…
Jim Brown
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What preserves similarity of invertible matrices?

Suppose $A$ and $B$ are invertible $n \times n$ matrices that are similar to each other. Then for example, $A - 2I$ and $B - 2I$ are similar, and $A^{-1}$ and $B^{-1}$ are similar. What other operations will preserve similarity, and what algebraic…
user901239012
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Extending linear function from a subspace to the whole (finite-dimensional) space

Let $V$ be a finite-dimensional vector space over the field $F$ and let $W$ be a subspace of $V$. If $f$ is a linear functional on $W$, prove that there is a linear functional g on $V$ such that $g(\alpha)$ = $f(\alpha)$ for each $\alpha$ in the…
ViKaN
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If $S$ spans a vector space $V$, does $S$ contain a basis?

As the title states, if $V$ is a vector space and $S \subseteq V$ spans $V$, does $S$ contain a basis? This is true if $V$ is finite dimensional. However, what if $V$ is infinite dimensional?
user136866
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Prove that S forms a subspace of R^3

Let S be the collection of vectors in $[x,y,z]$ in $R^3$ that satisfy the given property. In each case, either prove that $S$ forms a subspace of $R^3$ or give a counter example to show that it does not. Case: $z = 2x, \, y=0$ Okay,…
A A
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Proving that the range of linear transformation is a linear subspace

I need some help figuring out how to prove this question. True or false, the set $S = \left \{ A\mathbf{y}: \mathbf{y} \in \mathbb{R}^4\right \}$ is a subspace of $\mathbb{R}^3$ where A is a fixed $3\times4$ real matrix. Well I will need to show…
Bobby
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Linear transformations

Given $S$ and $T$, linear transformations such that $ST-TS$ commutes with $S$, I am supposed to show that $S^kT-TS^k=kS^{k-1} (ST-TS)$ for every positive integer $k$. As a matter of fact I do not know how to go about the solution. Can I have a…
smanoos
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In $P_2$, find the change-of-coordinates matrix

In $P_2$, find the change-of-coordinates matrix from the basis $B=\{1-2t+t^2, 3-5t+4t^2, 2t+3t^2\}$ to the standard basis $C=\{1, t, t^2\}$. Then find the B-coordinate vector for $-1+2t$ I know how to do the first part. $P$ from $B$ to $C:…
mmm
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The annihilator of an intersection

I know this question has been arlready asked, but as my reputation is too low I'm not allowed to post a comment, sorry for this second post. I'm asked to prove : $(W_1+W_2)^0=W_1^0\cap W_2^0$. $(W_1\cap W_2)^0=W_1^0+W_2^0$ I managed to prove the…
Jacques Ung
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Complex square matrix with distinct eigenvalues

Is there a simple way to show that if $A$ is a complex square matrix with distinct eigenvalues ​​then $A$ is similar to a matrix whose all entries are nonzero.
Mohamed
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Two subspaces of $\Bbb R^n$

If $A$ and $B$ are subspaces of $\mathbb R^n$. Is it possible to find a basis for $\mathbb R^n$ that contains a basis for $A$ and $B$? It has been suggested to me that we define a basis for $A\cap B$ and then use that to define basises $A$ and $B$.…
Mathman
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Can someone explain geometric multiplicity?

I'm reading my textbook and I'm really confused about geometric multiplicity. I've read the definition and they have given an example but I'm still lost. I've tried looking it up on other websites. This helped me make sense of algebraic multiplicity…
ayv2
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Proof that if the dimension of the complex space is n, then dimension of the real one is 2n

If a a set of vectors can be algebraized as an $\mathbb R$-vector space or a $\mathbb C$-vector, prove that if the dimension of the complex space is $n$, then dimension of the real one is $2n$. My idea was to try and prove it with the property…
Shomar
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Proving that $T^{2} = T$ for a linear operator on $W$ implies that $V = \operatorname{Null} T \oplus \operatorname{Range} T$

How can I prove that $T^{2} = T$ for a linear operator on $W$ implies that $V = Null T \oplus Range T?$ I know that their dimensions add up to the dimension of $W$, how do I show I can represent any element in $W$ like that? Or could I start with a…
hwlinal
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Basis for set of nxn matrices with trace = 0

I am trying to find a basis for the set of all $n \times n$ matrices with trace $0$. I know that part of that basis will be matrices with $1$ in only one entry and $0$ for all others for entries outside the diagonal, as they are not relevant. I…
nx__
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