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…)
Prove that a zero vector ${\bf 0}$ of a vector space $V$ is unique. (more…)
True or false:
a) Every vector space contains a zero vector;
b) A vector space can have more than one zero vector;
c) An $m\times n$ matrix has $m$ rows and $n$ columns;
d) If $f$ and $g$ are polynomials of degree $n$, then $f+g$ is also a polynomial of degree $n$;
e) If $f$ and $g$ are polynomials of degree at most $n$, then $f+g$ is also a polynomial of degree at most $n$. (more…)
Which of the following sets (with natural addition and multiplication by a scalar) are vector spaces. Justify your answer.
a) The set of all continuous functions on the interval $[0, 1]$;
b) The set of all non-negative functions on the interval $[0, 1]$;
c) The set of all polynomials of degree exactly $n$;
d) The set of all symmetric $n\times n$ matrices, i.e. the set of matrices $A =(a_{j,k})^n_{j,k=1}$ such that $A^T = A$. (more…)
Let ${\bf x} = (1,2,3)^T$, ${\bf y} = (y_1,y_2,y_3)^T$, ${\bf z}= (4,2,1)^T$. Compute $2{\bf x}$, $3{\bf y}$, ${\bf x}+2{\bf y}-3{\bf z}$. (more…)