True or false a Every vector space contains a zero vector b

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 × 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

Solution

a) Every vector space contains a zero vector= True

The existence of 0 is a requirement in the definition.

b) A vector space can have more than one zero vector =False.

That’s not an axiom, but you can prove it from the axioms. Suppose that z acts like a zero vector, that is to say, v + z = v for every vector v. Then in particular, 0 + z = 0. But z = 0 +z. Therefore, z = 0. Thus there can be only one vector with the properties of a zero vector.

c) An m × n matrix has m rows and n columns= True

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The dimensions of the matrix below are 2 × 3 (read \"two by three\"), because there are two rows and three columns.

d) If f and g are polynomials of degree n, then f + g is also a polynomial of degree n= False

The sum will have a lower degree if the leading coefficient of g is the negation of the leading coefficient of f.

e) If f and g are polynomials of degree at most n, then f +g is also a polynomial of degree at most n