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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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Divisibility of a Determinant by $2^{n-1}.$

Suppose $A$ is a $n \times n$ matrix with $0-1$ entries. Prove that the determinant of $X = nJ+B+B^*+2AA^T$ is divisible by $2^{n-1}$ where $B=(i-1)AJ$ and $J$ is the all ones matrix. I've reduced a tricky tournament problem to this simpler problem,…
Display name
  • 5,144
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Question in proof that $\dim(U_1+U_2)=\dim U_1+\dim U_2-\dim(U_1 \cap U_2)$

In the proof that Dim$(U_1+U_2)=DimU_1+DimU_2-Dim(U_1 \cap U_2)$ In "Linear Algebra Done Right" by Sheldon Axler: Theorem 2.18: If $U_1$ and $U_2$ are subspaces of a finite dimensional vector space, …
user736276
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anti symmetric matrix with generalized eigenspace not equal to the eigenspace

I would like an example of an anti symmetric matrix for which the $\lambda$-eigenspace is a strict subset of the generalized $\lambda$-eigenspace. If no such anti symmetric matrix exists I would like a proof of this. All of this is over the field C.…
Mathew
  • 1,894
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Existence of linear operator for given subspaces of vector space

Let $V$ be a finite-dimensional vector space and $L_1$ and $L_2$ are subspaces of $V$ such that $\dim L_1+\dim L_2=\dim V$. Show that exists linear operator $f$ such that, $$\ker(f)=L_1,\quad \text{Im}(f)=L_2.$$ My approach: Since $L_1$ is a…
RFZ
  • 16,814
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Trying to find a formula to calculate percentage reduction required per month

I've been messing around with compound interest calculations, but can't quite get this one right. I have a starting January value: $416.7$ I would like to reduce that value by the same percent "P" each month throughout the year from Jan thru Dec, so…
LarryBud
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Check if Linear Independence Proof is Accurate

.I am trying to prove the following statement (Linear Algebra Done Right, Section 2.A, #17): Suppose $p_0,p_1,\ldots,p_m$ are polynomials in $P_m(F)$ such that $p_j(2)=0$ for each $j.$ Prove that $p_0,p_1,\ldots,p_m$ is not linearly independent in…
user523642
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Direct sum problem linear algebra done right

Very confused on how to deal with these direct sum problems. Problem:Suppose $U=\{(x,y,x+y,x-y,2x) \in \mathbb{F}^{5}:x,y \in \mathbb{F}\}$ Find a subspace $W$ of $\mathbb{F}^{5}$ such that $\mathbb{F}^{5}=U \oplus W$ Trying to figure out a routine…
user736276
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True of False: For any $I_n$ and any $A \in \Bbb R^{n\times n}$, $(I_n + A)(I_n − A) = I_n − A^2$.

Not sure if the following statement is true or false: For any $I_n$ and any $A \in \Bbb R^{n\times n}$, $(I_n + A)(I_n − A) = I_n − A^2$ I am thinking true because: $(I_n + A)(I_n - A) = I_n*I_n - I_n*A + A*I_n - A*A = I_n - A + A - A^2 = I_n -…
Pranay Agrawal
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A linear operator of rank $1$ is diagonalizable or nilpotent but not both.

$T$ is a linear operator above field $F$. V is of finite dimension and $dim Im T = 1$; show that $T$ is diagonalizable or nilpotent but not both. This seems like a rather strong statement. I know that $T$ is diagonalizable if there is a basis $B$ of…
GoodWilly
  • 905
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Finding vectors which satisfy the cyclic decomposition theorem .

Let $f: \Bbb{R}^3 \to \Bbb{R}^3$ be a linear operator defined by $$f(x,y,z)=(3x-4y-4z,-x+3y+2z,2x-4y-3z)$$ Find non-zero vectors $v_1, \ldots,v_n$ satisfying the cyclic decomposition theorem.(i.e $\Bbb{R}^3=W_0 \bigoplus z(v_1,f) \bigoplus \ldots…
user63697
  • 597
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Use quaternion in place of rotation matrix

Given a vector n1, I find u = n1 x {0,1,0}. Then I find theta = acos(n1 . {0,1,0}). Then I form an axis-angle rotation matrix from u and theta. Then I form a rotation quaternion from that rotation matrix. My question is: Can I easily find n1 given…
TheBigO
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How can I say that this matrix is invertible?

Let $A_{ij}=1$ if $i=j$ and let $A_{ij}=\frac{1}{2}$ if $i \neq j$. How can I formally argue that $A$ is an invertible matrix? I can made for particular examples, but I don't know how to argue for the general case.
HeMan
  • 3,119
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Finding the eigenvalues of $\begin{pmatrix} a & b \\ b & a \end{pmatrix}$ matrix without the determinant

So we're reading Axler's Linear Algebra in class and haven't been taught the determinant. But we're asked to find the eigenvalues of $$\begin{pmatrix} a & b \\ b & a \end{pmatrix}$$ How can I go about doing this? I tried finding something that…
user_hello1
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On the existence a symmetric positive definite matrix

Let $u,v\in\mathbb{R}^n$ be such that $\langle u,v\rangle>0.$ My question is whether or not there exists a symmetric positive definite matrix $Q$ such that $v=Qu$. If such a matrix $Q$ exists, how to construct $Q$ from $u,v$? Thank you for all…
Blind
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What's the benefit of solving a diagonally dominant matrix compared to an ordinary one?

I have two linear systems of equations. One is strictly diagonal dominant and other is just an ordinary matrix. Both of them could have a very large scale. I'm wondering the benefit of solving a strictly diagonally dominant matrix compared to an…
Tony Dong
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