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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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Possible dimensions of the intersection of two subspaces

If $U$ and $W$ are subspaces of $V$ whose dimension is $9$, and $\dim(U) = 3$, and $\dim(W) = 5$, what could be the possible values of $\dim(U \cap W)$? By thinking about it it seems the possible values are $0, 1, 2, 3$ because the intersection…
nx__
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Is it true that $\det(A + I) = \operatorname{trace} (A) + 1$?

Let $A$ be an $n\times n$ matrix such that $\operatorname{rank} A = 1$ and that $n- 2$ rows of $A$ are the zero rows. Is it true that $\det(A + I) = \operatorname{trace} (A) + 1$? I already have the answer which can be viewed here. I don't…
GinKin
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Let $V$ be a finite dimensional vector space and $T$ be linear operator on $V$

Let $V$ be a finite dimensional vector space and $T$ be linear operator on $V$ such that $rank(T)=rank(T^2)$.Then to prove that the null space and range space of $T $are disjoint, i.e. zero vector is common.
user123804
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For what values of $x$ does $A^{-1}$ exist?

Let $A = \begin{bmatrix}4&-1&2\\5&x&7\\x&-1&3\end{bmatrix}$ I'm trying to find the values of $x$ such that $A^{-1}$ exists. What I have tried: $4(3x + 7) - (-1)(15)(7x) + (2)(-5)(x^2) = 0$ $-10x^2 + 117x + 28 = 0$ but I don't think that's right!
Cheryl
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Why are linear maps constructed to take on arbitrary values on a basis?

There is a section in Sheldon Axler's Linear Algebra Done Right (pg.40), where it says: "Linear maps can be constructed that take on arbitrary values on a basis. Specifically, given a basis $(v_1,...,v_n)$ of V and any choice of vectors $w_1,...,w_n…
user123276
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Necessary conditions for a matrix to have orthonormal eigenvectors?

A symmetric matrix has orthonormal eigenvectors and real eigenvalues. However, there are many kinds of matrices that have orthonormal eigenvectors and complex eigenvalues (for instance, circulant matrices). What are the necessary conditions for a…
David P
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Units of eigenvalues

Suppose you have the system $\bf x' = \bf Ax$, where $\bf x$ is a vector and $\bf A$ is a matrix. What are the units of the eigenvalues of $\bf A$? I think they should be $1/t$ but I'm not sure how to verify this. Can you give me a starting point?
liarose
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Variable Tikhonov Parameter

In the Tikhonov regularization problem, $\Vert Ax-b\Vert^{2}+\Vert\Gamma x\Vert^{2}$, with $\Gamma=\alpha I$ .The solution from SVD is $x=VDU^\top$, where $A=U\Sigma V^\top$ and $D_{ii}=\dfrac{\sigma_i}{\sigma_i^2+\alpha^2}$, with $\sigma_i$ given…
user16409
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$tf_1,t^2f_2,t^3f_3$ are linearly indipendent in $C[0,1]$ does this imply $f_1,f_2,f_3$ are linearly indipendent?

Given that $tf_1,t^2f_2,t^3f_3$ are linearly indipendent in $C[0,1]$ does this imply $f_1,f_2,f_3$ are linearly indipendent? $t\to t^n$ is a polynomial, $f_i\in C[0,1]$ Thank you for helping.
Myshkin
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Show that minimal polynomial for $n\times n$ matrix and its transpose is the same

Show that minimal polynomial for $n\times n$ matrix and its transpose is the same
user124994
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Intuition behind this theorem in linear algebra

I can't get my head around this introductory theorem from Lang's Linear Algebra. Let $V$ and $W$ be vector spaces and $\{ v_1,\cdots v_n\}$ be a basis of $V$ and $w_1,\cdots , w_n$ be arbitrary elements of $W$. Then there exists a unique mapping…
kuch nahi
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Find a real $2\times 2$ matrix $A$ (other than $A = I$ ) such that $A^5 = I $.

I found the question in an online a source of challenging linear algebra problems, unfortunately there are no answers. Question: Find a real $2\times 2$ matrix $A$ (other than $A = I$ ) such that $A^5 = I$. I'm beginning to think no such matrix…
Grid
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Finding a simple basis of a subspace of $R^{N}$

Say I have some basis of a subspace in $R^{N}$. I would like to find a new basis that is as simple as possible, meaning one that contains vectors that have as many zero elements as possible. I expect that I need some kind of matrix decomposition to…
RobVerheyen
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Solving differential equations in linear algebra

I'm having a hard time early on in this linear algebra course, I'm a first year student in University. I'm reading my textbook right now and it gives the following differential equation as an example with a solution and I still can't understand how…
Sam Creamer
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Prove: a set of vectors $K$ is linearly dependent iff a vector is linear combination of the others

Prove: a set of vectors $K$ is linearly dependent iff a vector is linear combination of the others. Let: $\alpha_1k_1 + \alpha_2k_2+...\alpha_nk_n = 0$ Then, There must be $\alpha_i \ne 0$. Therefore, $$\alpha_1k_1 + \alpha_2k_2 +…
SuperStamp
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