Corner : A corner of a shape is a point in the shape such that this point does not lie on the line segment between any two other points in the shape. Every triangle has 3 corners and every quadrilateral has 4 corners.

Intersect : Two shapes intersect if they have at least one point in common.

Line segment : A line segment AB is a one-dimensional shape consisting of the points A and B and all the points on the straight connection of A and B. A and B are both endpoints of the line segment AB.

Midpoint : The midpoint of a line segment is the point on the line segment that cuts the line segment into two equal length pieces.

Point : A point is an element in a space. Shapes are made of sets of points.

Ratio : A ratio is a comparison of two numbers using a division.

Right triangle : A right triangle is a triangle that has a right angle. The sides that form the right angle are called legs, the third side is called hypotenuse. The area of a right triangle is half the product of the lengths of the two legs. The Pythagorean theorem is an important theorem for right triangles that allows to calculate the third side of a right triangle given the other two.

Triangle : A triangle is a polygon with three corners and three sides. You can calculate the area of a triangle by multiplying half the length of the base by the height on that base. The sum of the interior angles in a triangle is always $180^{\circ}.$

Inequality : An inequality is a statement that contains an inequality sign between two expressions.

Length : Length is the attribute of a one-dimensional shape that can be measured with a measuring tape.

Norm : The norm $\lVert v\rVert$ of a vector $v=\begin{pmatrix}v_1\\v_2\\ \vdots \\v_n\\\end{pmatrix}$ is given by $\lVert v\rVert=\sqrt{v\cdot v}$ $=\sqrt{v_1^2+v_2^2+\ldots+v_n^2}.$ The norm of a vector intuitively describes the length of that vector.

Real number : A real number is any number that can be represented by a possibly infinite decimal expansion. All rational and irrational numbers are real numbers.

Sum : A sum is the result of an addition.

Third : A third either refers to the third object in an ordering or to the number $\frac{1}{3}=0.\overline{3}.$

Triangle : A triangle is a polygon with three corners and three sides. You can calculate the area of a triangle by multiplying half the length of the base by the height on that base. The sum of the interior angles in a triangle is always $180^{\circ}.$

Triangle inequality : According to the triangle inequality we have $\lVert x+y\rVert\leq\lVert x\rVert+\lVert y\rVert$ for all vectors $x,y$ in $\mathbb{R}^{n}.$ It is called triangle inequality as it can be interpreted as saying that any one side of a triangle is always shorter than the other two sides combined.

Vector : Most commonly vector refers to a matrix with one column $\begin{pmatrix}x_1\\ \vdots \\x_n\\\end{pmatrix}.$ In general a vector is an element of a vector space.

Basis : The basis of a vector space is a set of linearly independent vectors, so that every vector of the vector space can be written as a linear combination of the vectors in the basis. For example $\{\left(\begin{smallmatrix}1\\0\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\1\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\0\\1\\\end{smallmatrix}\right)\}$ is a basis of $\mathbb{R}^3.$ In a prove by induction basis refers to the first step in which you have to prove that the result is true for a particular number $n$ (usually $n=0$ or $n=1$.)

Independence : Random variables $X,$ $Y$ are called independent if $P[X\in A, Y\in B]=P[X \in A]P[Y \in B].$ Independent identically distributed random variables feature prominently in the law of large numbers and the central limit theorem. Vectors $x_1,x_2\ldots, x_n$ are called linearly independent if $\lambda_1 x_1+\lambda_2 x_2+\ldots+\lambda_n x_n=0$ implies $\lambda_1=\lambda_2=\ldots=\lambda_n=0.$

Linear independence : Vectors $x_1, x_2,\ldots, x_n$ in a vector space are called linearly independent if $\lambda_1 x_1+\lambda_2x_2+\ldots+\lambda_nx_n=0$ implies $\lambda_1=\lambda_2=\ldots=\lambda_n=0.$ In other words vectors are linearly independent if the only linear combination of the zero vector is the trivial one in which all coefficients are zero.

Three-dimensional coordinate vector space : The vector space of all three-dimensional vectors $\begin{pmatrix}a\\b\\c\\\end{pmatrix}.$

Vector : Most commonly vector refers to a matrix with one column $\begin{pmatrix}x_1\\ \vdots \\x_n\\\end{pmatrix}.$ In general a vector is an element of a vector space.