# Questions tagged [linear-algebra]

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

82,560 questions
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### Using the product of the roots of $(z+1)^n=1$ to prove that $\prod_{k=1}^{n-1} \sin\frac{k\pi}{n}=\frac{n}{2^{n-1}}$

$z_1,z_2,…z_n$ satisfy the equation $(z+1)^n=1$. Use the product of $z_1,z_2,…z_n$ to prove that $$\sin \frac {\pi}{n} \sin \frac {2\pi}{n}…\sin \frac {(n-1)\pi}{n}=\frac {n}{2^{n-1}}$$ Attempt I ...
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### Linear independence of gradient vectors

I have to find all feasible points that are regular for a function with constraints: $h_1(x_1,x_2,x_3) = 2x_1x_2+x_3^2=0$ $h_2(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2=4$ The definition says that if a ...
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### Eigenvalues and Eigenvectors of special matrix $A=u u^T$

How to calculate the eigenvalues and eigenvectors of the special matrix $A=u u^T$, where $u \in R^n$. I wrote down the matrix, which is the linear combination of vector $u$ by itself, and clearly has ...
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### When does the base of the image is directly comprized of the columns of a matrix?

I am confused with what methods and when to use them to find the base of the image of a matrix. Sometimes I see that they use gauss-jordan to find which columns have pivots and then they take the ...
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### How $C[a, b]$ satisfies the axioms of vector space

My book states that the set of all continuous functions defined on a closed interval $[a, b]$ where $a ≠ b$ satisfies the axioms of a vector space. I am understanding that this means that for the ...
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### Linear algebra identity evaluation

I really couldn't find anything related to this simple identity I came up with so: $$\vec{r}=(r_x,r_y)=(r_x, \angle0)+(r_y,\angle\frac{\pi}{2})$$ My thinking process was that $r_y$ is practically ...
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### Groups, Rings and Fields.

I am asking for the analogy behind these structures names. Why is a "field" called a field? Is there an analogy between a usual ring (finger ring) and a mathematical ring?
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### Gramian matrix after elementary matrix operations

Will a Gramian matrix remain a Gramian matrix after elementary matrix operations(for example, subtract a row from another row and similarly to columns)? Of course, we do this with symmetry about ...
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### Distance between two points at same angle in trochoid curve

Anyone please help me to find out the distance in following case. Refer to the attached image. Consider an arbitrary point P on the circumference of a circle of radius r (mm). The point makes an ...
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### Determining all possible ways to perform gaussian elimination on a linear system of equations

My first question is quite general: consider a linear system \begin{equation} A\vec{x}=\vec{b}, \end{equation} with $A\in\{-1,0,1\}^{n^2}$ being a full rank matrix and $\vec{x},\vec{b}\in\mathbb{R}^n$...
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### How to prove that a cofactor of a matrix is $A$ is $(-1)^{i+j} \times$ a minor

Let $A \in M_n(\mathbb{R})$ : $A'_{ij}$ have the same columns as $A$ except the $j$th one which is a column full of zero except on the $i$th entry where it is a $1$. $A''_{ij}$ is the matrix ...
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### Eigenvalues of the sum of commuting matrices

In a proof I am reading, the author states that given any commuting matrices $A$ and $B$, any eigenvalue of $A+B$ must be the sum of an eigenvalue of $A$ and an eigenvalue of $B$. Why is this true? Is ...
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### The orthogonal complement of the orthogonal complement from “Linear Algebra Done Right”

The following content is from "Linear Algebra Done Right" by Sheldon Axler Corollary: Suppose $U$ is a finite-dimensional subspace of $V$. Then $$U = (U ^\perp)^\perp.$$ We need to prove the ...
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### Merging two orthonormal bases without Gram-Schmidt

I have two sets of column vectors: $A = \{a_1,a_2,\dotsc,a_m\}$ and $B = \{b_1,b_2,\dotsc,b_n\}$. I have orthornormal basis for both of them individiaully. $\{u_1,\dotsc,u_p\}$ is an othornormal basis ...
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### nilpotent endomorphism and $Im(f)+Ker(f) \neq dim(V)$

If I have an endomorphism between vector spaces $f:V \rightarrow V$, such that $Im(f)+Ker(f) \neq dim(V)$, is this equivalent to $f$ being nilpotent?
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### How is the definition of linear independence for infinite sets useful?

Given this fact: "an infinite set is linearly independent iff all of its finite subsets are linearly independent", how can this help us determine whether an infinite set is linearly independent if ...
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### How does one find a parameter representation for bounded region?

I need help with this question. I have been stuck at it for a few days. My main problem is how I use the curve $K_r$ to find the parametric representation. I have a curve $K_r$ in the $(x,y)$-plane ...
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### Similarities between the idea of Group Homomorphisms and Linear Transformations

I am not sure if this question has been asked before, but my search did not return me any answers. While reading several online notes that attempt to give an intuitive understanding of group ...
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### Proof that Irreducible Block Diagonally Dominant Matrix is Nonsingular

Let $A$ be an $n\times n$ block matrix with $ij$th block $A_{i,j}$, where each $A_{i,j}$ is a square matrix of the same size for all $i,j$. Assume that $A$ is block diagonally dominant: each of the ...
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### Existence of diagonalizable matrix

Let $A \in \mathbb{C}^{4 \times4}$ and $A^5 = I$. Is $A$ always diagonalizable? How to show $|\mbox{tr} (A)| \leq 4$? If $\mbox{tr} (A) = 4$, can we determine matrix $A$? Some ...
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### The definition of normal bundle as a quotient.

Let $f: N \to M$ be a smooth immersion and let $p \in M$, $W = f(V) \subset M$ be an submanifold with $q = f(p).$ Then the sequence is split exact T_qW \hookrightarrow T_qM \stackrel{\mu}\to T_qM/...
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### How can we generalize the fact of finite dimensional vector space to an infinte dimensional case?

I am reading vector space from Friedberg. There in the last section they told about infinite dimensional vector space but there is not sufficient contents. Now my question is why can't we define ...
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### Describe vector subspaces and FInd basis for sum of vector subspaces

I need help with this problem: For part a), I thought to write $v_1,v_2,v_3$ in their vector form with respect to the standard basis. Then I will group them together into a matrix and row reduce the ...