# Tagged Questions

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 ...

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### Origin of the dot and cross product?

Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...
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### How can I calculate the correct rotation output when input needs modification?

I am making a GPS for vehicles in a game and I need them to turn accordingly to the rotation between the ending point and starting point. I have calculated the rotation, and before I can apply the ...
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### Relation between infinity norm and LU factorization

Let $A$ be a non-singular $n \times n$ matrix and suppose that Gaussian elimination with partial pivoting has been applied to produce $PA = LU$, where $P$ is a permutation, $L$ is a unit lower ...
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### Strang , Introduction to Linear Algebra, Clarification of Breakdown of Elimination Chapter 2

I am sorry in advance if my question is very simple, I am just a beginner. In Strang's "Introduction to Linear Algebra" in Section 2.2 page # 46 , he is explaining "Breakdown of Elimination", and he ...
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### Prove that a Vector Orthogonal to an Orthonormal Basis is the Zero Vector.

Stuck on this proof. Let W be an inner product space (with unspecified inner product, $<\vec x, \vec y>$), and with orthonormal basis B = {$\vec w_1, \vec w_2, \ldots ,\vec w_n$}. Suppose ...
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### Find the determinant by inspection

I'm supposed to "use properties of determinants to evaluate the determinant by inspection" on this matrix: $$\pmatrix{ 0 & 0 & 3 \\ 0 & 4 & 1 \\ 2 & 3 & 1 }$$ I don't see ...
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### The transpose of a linear injection is surjective.

Let $$T:V\longrightarrow W$$ be a linear map (of vector spaces), and let \begin{eqnarray} T^*:W^* &\longrightarrow& V^* \\ f\ &\longmapsto& f\circ T \end{eqnarray} be its transpose ...
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### Why does $A \circ {B^{ - 1}} + {A^{ - 1}} \circ B \ge 2{I_{n \times n}}$?

Let $A, B \in M_n$ be positive definite and $A \circ B = \left[ {{a_{ij}}{b_{ij}}} \right]$. Why does $A \circ {B^{ - 1}} + {A^{ - 1}} \circ B \ge 2{I_{n \times n}}$ ?
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### How to find distance between vector and a subspace

Well, this is question from a test that I had, I didn't know how to answer it so I am forwarding this to you: Consider $v\:=\:\begin{pmatrix}\frac{1}{3} \\\frac{2}{3}\: \\\frac{2}{3}\end{pmatrix}$. ...
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### Find change of basis matrix

I'm asked to find the change of basis matrix from basis $\underline{e}$ to $\underline{f}$ given the following information: The coordinate relationship is given by: $$3y_1 = -x_1 + 4x_2 + x_3$$ ...
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### Square-root of a matrix which arise from truncating a matrix which has a square-root

I have this covariance matrix $A$ which has a symmetric Toeplitz structure. A = \left[ \begin{array}{cccccccc} c_0 & c_1 & c_2 & \cdots & c_{n-1} & c_{n} \\ c_1 ...
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### is it true that $\underline{\sigma}(I+AA^T)=\underline{\sigma}(I)+\underline{\sigma}(AA^T)$

is it true that $\underline{\sigma}(I+AA^T)=\underline{\sigma}(I)+\underline{\sigma}(AA^T)$ where $\underline{\sigma}$ denotes the smallest singular value.
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### Proofs in Linear Algebra via Topology

I'm watching the lecture series by Tadashi Tokieda on Topology and Geometry on YouTube. In the second lecture he shows how one can prove, using a topological argument, that given two $n\times n$ ...
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### Finding All Invariant Subspaces of Given Transformation

I'm making my way through Axler's Linear Algebra Done Right and I'm a bit stumped by this exercise. Any help would be appreciated. Define $T\in\mathcal{L}(\mathbb{F}^{n})$ (a linear transformation ...
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### How does $D=C^{-1}AC$ matrix of a linear map with respect to alternate basis.

I was watching a Khan video on the topic and wondered how you can use the graph he draws to derive $D=C^{-1}AC$ I see how it follows without using the graph but the graph confuses me. Looking at this ...
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### Prove that $q \leq \dim(E_\lambda)$ given that $J$ has $q$ Jordan blocks associated with $\lambda$

Let $T: V \rightarrow V$ be linear, $V$ is a finite dimensional vector space, and the characteristic polynomial of $T$ splits. Also let $\lambda$ be an eigenvalue of $T$ and $B$ be a Jordan Canonical ...
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### any $T\in CL(Z,Y)$ has a generalized inverse which is a topological homomorphism.

Corollary: If $Z$ and $Y$ are complete metrizable topological vector spaces, then any $T\in CL(Z,Y)$ such that $N(T)$ is complemented in $Z$ and $R(T)$ is complemented in $Y$ has a generalized ...