Solution to Linear Algebra Hoffman & Kunze Chapter 9.2 Exercise 9.2.2 Solution: It is not a form! I think it should be $$f((x_1,y_1),(x_2,y_2))=x_1x_2+y_1y_2.$$Then the matrices are\[\begin{pmatrix}1 & 0\\ 0 &…
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?
(more…)
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…)