# 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.

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### Definitions of the operator norm of a matrix

I understand that the operator norm of a matrix is induced by the vector norm, and is given by $$\|A\|_{\rm op} = \max_{\|x\| = 1}\|Ax\|$$ However, in a lecture, I see the operator norm being ...
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### Eigen vector of the adjoint operator. Error in this approach?

Reisz Representation Theorem: If $V$ is finite-dimensional and $\phi$ is a linear functional on $V$. Then, there is a unique vector $v \in V~|~\phi(v) = \langle v,u\rangle~\forall~v \in V$ Definition: ...
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### A matrix polynomial convergent to a matrices’ transpose

Is it true that for all $A \in \mbox{GL}(n)$ there exists $c_i$ such that the following holds? $$\sum^\infty_{i=0} c_i A^i = A^T$$ For symmetric matrices, it is simple. I suspect some rotation matrix ...
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### Why a subspace is assumed instead of a vector space?

Why does the following theorem start with "Let $\{u_1, ..., u_p\}$ be an orthogonal basis for a subspace $W$ of $\mathbb{R}^n$" instead of "Let $\{u_1, ..., u_p\}$ be an orthogonal ...
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### A question regarding matrix of inner product in the ordered basis from Hoffman Kunze

While studying Chapter - Inner Product Spaces from Hoffman Kunze, I have a question in section 8.1 . Adding it's image: How can I derive the condition that the matrix must satisfy the additional ...
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### Calculate real matrix inverse of a complex matrix

Given a Hermitian positive semidefinite matrix $A \in \mathbb{C}^{n \times n}$. If $B=A^{-1},\ D=\text{Re}(B),\ C=D^{-1}$. where $D=\text{Re}(B)\Leftrightarrow{}d_{ij}=\text{Re}(b_{ij})$. Can I ...
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### If $U$ and $T$ are one to one and onto then $UT$ is also - why is my proof not sound?

Let $V$, $W$, and $Z$ be vector spaces, and let $T: V \to W$ and $U: W \to Z$ be linear. Prove that if $U$ and $T$ are one to one and onto then $UT$ is also. I've seen other solutions to this problem ...
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### Quadratic form can be represented as a convex combination of $\frac{n(n+1)}{2}+1$ ones

Question : In $\mathbb{R}^n$, consider an inner product $(\ ,\ )$. Here any linear map $L$ has the form $L(x)=(V,\ x)$ for some $V$ When $\|\ \|,\ \|\ \|^\ast$ are norms in dual relation, then we have ...
### Show that $I$ is an ideal of $\mathbb{K}[x]$
Let $\mathbb{K}$ be a field, $x_1, x_2, x_3\in \mathbb{K}$ and $I:=\left \{f\in \mathbb{K}[x]\mid f(x_1)=f(x_2)=f(x_3)=0\right \}$. I want to show that $I$ is an ideal of $\mathbb{K}[x]$. So we take \$...