Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.7 Let $F$ be a field and $V$ an $F$-vector space. The axioms for vector spaces over…
Let a system of vectors ${\bf v}_1$, ${\bf v}_2$, $\cdots$, ${\bf v}_r$ be linearly independent but not generating. Show that it is possible to find a vector ${\bf v}_{r+1}$ such that the system ${\bf v}_1$, ${\bf v}_2$, $\cdots$, ${\bf v}_r$, ${\bf v}_{r+1}$ is linearly independent. Hint: Take for ${\bf v}_{r+1}$ any vector that cannot be represented as a linear combination $\sum_{k=1}\alpha_k{\bf v}_{k}$ and show that the system ${\bf v}_1$, ${\bf v}_2$, $\cdots$, ${\bf v}_r$, ${\bf v}_{r+1}$ is linearly independent. (more…)
Write down a basis for the space of
a) $3\times 3$ symmetric matrices;
b) $n\times n$ symmetric matrices;
c) $n\times n$ anti-symmetric ($A^T = -A$) matrices; (more…)
Recall, that a matrix is called symmetric if $A^T = A$. Write down a basis in the space of symmetric $2\times 2$ matrices (there are many possible answers). How many elements are in the basis?
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True or false:
a) Any set containing a zero vector is linearly dependent;
b) A basis must contain ${\bf 0}$;
c) subsets of linearly dependent sets are linearly dependent;
d) subsets of linearly independent sets are linearly independent;
e) If $\alpha_1{\bf v}_1+\alpha_2{\bf v}_2+\cdots+\alpha_n{\bf v}_n=0$ then all scalars $\alpha_k$ are zero. (more…)
Find a basis in the space of $3\times 2$ matrices $M_{3\times 2}$. (more…)
Prove that for any vector ${\bf v}$ its additive inverse $−{\bf v}$ is given by $(−1){\bf v}$. (more…)
Prove that $0{\bf v}={\bf 0}$ for any vector ${\bf v}\in V$. (more…)
Prove that the additive inverse, defined in Axiom 4 of a vector space is unique. (more…)
What matrix is the zero vector of the space $M_{2\times 3}$? (more…)