## 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 $(−1){\bf v}$. (more…)

This author has written 89 articles

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…)

Chapter 1. Basic Notions Vector spaces #1.1, #1.2, #1.3, #1.4, #1.5, #1.6, #1.7, #1.8 Linear combinations, bases #2.1, #2.2, #2.3, #2.4, #2.5, #2.6 Linear Transformations. Matrix–vector multiplication #3.1, #3.2, #3.3, #3.4, #3.5, #3.6, #3.7 Linear transformations as…