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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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Does linearly independent imply all elements are orthogonal?

As the title states, if you have $A\subset V$ where $V$ is a vector space over an arbitrary field, does $A$ being linearly independent imply that the elements of $A$ are orthogonal?
Complimentarity
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Find a homogeneous system whose solution space is spanned by the given vectors

find a homogeneous system whose solution space is spanned by the following set of 3 vectors: $$(1,-2,0,3,-1) , (2,-3,2,5,-3), (1,-2,1,2,-2)$$ Please help, I've only seen similar questions where there are 4 unknowns not 5
Pheobe
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Is there a formula for the inverse of this bordered matrix?

Suppose I have a matrix $\mathbf{H}$ of size $n\times n$, and that I know its inverse $\mathbf{W}=\mathbf{H}^{-1}$. Then I add a column and a row to $\mathbf{H}$ to obtain a new matrix $\mathbf{G}$. That is $\mathbf{G}$ is given…
npisinp
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Linearly independent functions

Show that the set consisting of the functions $$x, e^x, e^{-x}$$ on $\mathbb R$ is linearly independent. So I have the equation $$ax + be^x + ce^{-x} = 0$$ and I want to show that this is only satisfied when $a = b = c = 0$ Letting x = 0, $b + c =…
Jim_CS
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Determine the null space of a linear map

Let $P_k(x)$ denote the space of polynomials of at most degree $k$. Let $D$ denote differentiation with respect to $x$. Regard the differential operator $L: P_k\rightarrow P_k$ such that $L=\frac{1}{n!}D^n+\frac{1}{(n-1)!}D^{n-1}+...+D+I$ . If…
nerd
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classification up to similarity of complex n-by-n matrices

Classify up to similarity all 3 x 3 complex matrices $A$ such that $A^n$ = $I$.
Josh
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Understanding Householder Transformations

I've been thinking about Householder transformation for the past few days and one point appears to be escaping my insight. I hope that the following description helps someone correct my understanding and point me in the right direction. Based on a…
ariadnus
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Finding a fourth vector that makes a set a basis

The following vectors are linearly independent - $v1 = (1, 2, 0, 2)$ $v2 = (1,1,1,0)$ $v3 = (2,0,1,3)$ Find a fourth vector v4 so that the set { v1, v2, v3, v4 } is a basis fpr $\mathbb{R}^4$? I asked this question before here - Show vectors are…
Jim_CS
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Are there nontrivial vector spaces with finitely many elements?

I have only seen infinite vector spaces and the one finite vector space i.e the trivial vector space $\{0\}$. Is there any other finite vector space?
Waqar Ali Shah
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question on self adjoint operator

Suppose $A$ is a $n\times n$ matrix with complex entries and $A^*A=A^2$. Does it imply $A=A^*.$
gopal dutta
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Why does a symmetric matrix have a complete set of eigenvectors and eigenvalues?

I am attempting to learn more about the adjacency matrix(graph theory) but given that I have forgotten a lot of linear algebra, I can't seem to know why this is true. Can someone give me a proof?
tintinthong
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Kernel of a matrix pencil

Let $A,B$ be $n\times n$ singular real matrices such $ker A\cap ker B=\{0\}$, how could I show that there exists $x\in \mathbb R$ such that $ker (A+xB)=\{0\}$?
user203039
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Inverse of a matrix having zeroes in diagonal and one elsewhere

Could any one help me to find inverse of such matrix? I observed that $A= J-I$, where J is a matrix having all entries 1. Thanks for helping.
Myshkin
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Dimension of vector space of real numbers over rational number field

I know that dimension of $\mathbb{R} $ over $ \mathbb{Q} $ is infinite. What can i say about the cardinality of its basis mean whether it is countable or uncountable. Can we find exact basis for that.
user195218
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If $T\alpha=c\alpha$, then there is a non-zero linear functional $f$ on $V$ such that $T^{t}f=cf$

I am self-studying Hoffman and Kunze's book Linear Algebra. This is exercise $4$ from page $115$. It is in the section of The transpose of a Linear Transformation. Let $V$ be a finite-dimensional vector space over the field $\mathbb{F}$ and let…
user23505
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