Vector space14

The action of the nonzero elements of a field on a vector space by left multiplication forms a faithful group action
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.7 Let $F$ be...
Solution to Linear Algebra Done Wrong Exercise 1.2.5
Let a system of vectors ${\bf v}_1$, ${\bf v}_2$, $\cdots$, ${\bf v}_r$ be linearly independent but ...
Solution to Linear Algebra Done Wrong Exercise 1.2.4
Write down a basis for the space of a) $3\times 3$ symmetric matrices; b) $n\times n$ symmetric matr...
Solution to Linear Algebra Done Wrong Exercise 1.2.3
Recall, that a matrix is called symmetric if $A^T = A$. Write down a basis in the space of symmetric...
Solution to Linear Algebra Done Wrong Exercise 1.2.2
True or false: a) Any set containing a zero vector is linearly dependent; b) A basis must contain ${...
Solution to Linear Algebra Done Wrong Exercise 1.2.1
Find a basis in the space of $3\times 2$ matrices $M_{3\times 2}$.
Solution to Linear Algebra Done Wrong Exercise 1.1.8
Prove that for any vector ${\bf v}$ its additive inverse $−{\bf v}$ is given by $(−1){\bf v}$.
Solution to Linear Algebra Done Wrong Exercise 1.1.7
Prove that $0{\bf v}={\bf 0}$ for any vector ${\bf v}\in V$.
Solution to Linear Algebra Done Wrong Exercise 1.1.6
Prove that the additive inverse, defined in Axiom 4 of a vector space is unique.
Solution to Linear Algebra Done Wrong Exercise 1.1.5
What matrix is the zero vector of the space $M_{2\times 3}$?