Question

Testing for a Vector Space In Exercises 13-36, determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. 13. $M_{4,6}$ 14. $M_{1,1}$ 15. The set of all third-degree polynomials 16. The set of all fifth-degree polynomials 17. The set of all first-degree polynomial functions $ax$, $a \neq 0$, whose graphs pass through the origin 18. The set of all first-degree polynomial functions $ax + b$ 30. The set of all $4 \times 4$ matrices of the form $\begin{bmatrix} 0 & a & b & c \\ a & 0 & b & c \\ a & b & 0 & c \\ a & b & c & 0 \end{bmatrix}$ 31. The set of all $2 \times 2$ singular matrices 32 The set of all $2 \times 2$ nonsingular matrices 33. The set of all $2 \times 2$ diagonal matrices 34. The set of all $3 \times 3$ upper triangular matrices 35. $C[0, 1]$, the set of all continuous functions

          Testing for a Vector Space In Exercises 13-36,
determine whether the set, together with the standard
operations, is a vector space. If it is not, identify at least
one of the ten vector space axioms that fails.
13. $M_{4,6}$
14. $M_{1,1}$
15. The set of all third-degree polynomials
16. The set of all fifth-degree polynomials
17. The set of all first-degree polynomial functions $ax$,
$a \neq 0$, whose graphs pass through the origin
18. The set of all first-degree polynomial functions $ax + b$
30. The set of all $4 \times 4$ matrices of the form
$\begin{bmatrix} 0 & a & b & c \\ a & 0 & b & c \\ a & b & 0 & c \\ a & b & c & 0 \end{bmatrix}$
31. The set of all $2 \times 2$ singular matrices
32 The set of all $2 \times 2$ nonsingular matrices
33. The set of all $2 \times 2$ diagonal matrices
34. The set of all $3 \times 3$ upper triangular matrices
35. $C[0, 1]$, the set of all continuous functions

Testing for a Vector Space In Exercises 13-36,
determine whether the set, together with the standard
operations, is a vector space. If it is not, identify at least
one of the ten vector space axioms that fails.
13. M4,6
14. M1,1
15. The set of all third-degree polynomials
16. The set of all fifth-degree polynomials
17. The set of all first-degree polynomial functions ax,
a ≠ 0, whose graphs pass through the origin
18. The set of all first-degree polynomial functions ax + b
30. The set of all 4 × 4 matrices of the form
< b m a t r i x >
31. The set of all 2 × 2 singular matrices
32 The set of all 2 × 2 nonsingular matrices
33. The set of all 2 × 2 diagonal matrices
34. The set of all 3 × 3 upper triangular matrices
35. C[0, 1], the set of all continuous functions

Added by Linda B.

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