# Questions tagged [linear-algebra]

For questions about 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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### $A = B+ C$ then $(B^{-1} - A^{-1})$ and $(C^{-1} - A^{-1})$ are positive semidefinite

Let $A$, $B$ and $C$ invertible and symmetric square real matrices of dimension $n$. I want to show that if $A = B+ C$ then $(B^{-1} - A^{-1})$ and $(C^{-1} - A^{-1})$ are positive semidefinite. For a ...
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### Linear Transformation Linearity

This might be very straightforward, but if $S,T,R$ are all linear transformations in vector space V, is $(ST+RT) = (S+R)T$
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### How is this type of matrix called?

Let the matrix $M$ be defined as: $$M = \left(\begin{array}{rr} A & B \\ C & D \end{array}\right)$$ where $A = 0,B,C,D = 0$ are block matices and $B$ is a diagonal matrix and $C$ is a diagonal ...
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### Let $S_1$ and $S_2$ be convex sets. Show that the intersection $S_1 \cap S_2$ is a convex set

Found this exercise in "Introduction to Linear Algebra" by Serge Lang. It says: "Let $S_1$ and $S_2$ be convex sets. Show that the intersection $S_1 \cap S_2$ is a convex set" My ...
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### Need help with proof of [closed]

Let {~v1, · · · ~vm} ⊂ Rn be a linearly independent set. Let L ∈ Rm×n . Prove that, if null(L) = {~0}, then {L~v1, · · · L~vm} ⊂ Rm is a linearly independent set.
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### Sets that contain $2$ or $1$ different elements can we make them vector spaces?

Sets that contain $2$ or $1$ different elements can we make them vector spaces? Answer is for $1$ element $yes$ for $2$ element $no$. I know definition of vector space and it's $8$ axioms but don't ...
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### Prove that if AB=I_m, then rk(A)=rk(B). [closed]

If $A\in\mathfrak{M}_{m,n}(F)$ and $B\in\mathfrak{M}_{n,m}(F)$, prove that $AB=I_m$ implies $rk(A)=rk(B)$. Here $A\in\mathfrak{M}_{m,n}(F)$ means $A$ is a ($m \times n$)-matrix with entries in $F$. My ...
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### The value/s of $a$ will the linear system have a unique, infinite and no solution?

For which value/s of $a$ will the linear system have a unique, infinite, and no solution? $\begin{cases} x+y-z=2\\ x+2y+z=3 \\ x+y+(a^2-5)z=a\end{cases}$ Can someone verify my solution? Thank you. My ...
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### Related to Euclidean and unitary vector space

Let $V$ be a finite-dimensional Euclidean or unitary $K$-vector space. Show or refute the following statements: (i) $(f + g)^{*} = f^{*} + g^{*}$ for all $f, g ∈ \operatorname{End}(V)$ (ii) $(λf)^{*}$ ...
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### prove that the set of all differentiable functions on (−∞, ∞) that satisfy f ′ + 2f = 0 is a subspace of F(−∞, ∞) [closed]

How can I prove that the set of all differentiable functions on (−∞, ∞) that satisfy f ′ + 2f = 0 is a subspace of F(−∞, ∞)
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### Minimizing the norm of the difference of two vectors.

Let $\mathcal{H}$ be a a vector space of finite dimension. Let $\mathcal{H_1}$ be a subspace of $\mathcal{H}$. Considering some vector $|\phi\rangle \in \mathcal{H}$ i need to show that there exists ...
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### On the linear dependence of three coplanar vectors

The general consensus seems to be that any three coplanar vectors are linearly dependent. Here's one source that says so. However, considering three vectors, of which two are collinear and the third ...