Given information
We are given with some statements, we have to determine whether the statement is True or False.

Step 1: (a)
Statement: The equation $A{\bf{x}} = b$ is referred to as a vector equation.

The Statement is False. In the above form, the entities $x$ and $b$ are vectors. However the entity, A is a matrix. hence the given form is a matrix form.
Therefore,
FALSE

Step 2: (b)
Statement: A vector b is a linear combination of the columns of a matrix A if and only if the equation $A{\bf{x}} = b$ has at least one solution..

The Statement is True. If the equation $A{\bf{x}} = b$ has at least one solution, then the vector be can be written as linear combination of vectors of matrix A, with coefficient as solution set ($x$).
Therefore,
TRUE

Step 3: (c)
Statement: The equation $A{\bf{x}} = b$ is consistent if the augmented matrix [A b] has a pivot position in every row.

The Statement is False. The above statement is true for the coefficient matrix ($A$) but not for the augmented matrix. Even if the augmented matrix has a pivot element in every row, the solution may or may not be consistent.
Example. \[M = \left[ {\begin{array}{*{20}{c}}1&0&{32}\\0&0&1\\0&1&3\end{array}} \right]\]The second row can not be true, even though every row has a pivot element. Therefore,
FALSE

Step 4: (d)
Statement: The first entry in the product Ax is a sum of products.

The Statement is True. The $i^th$ entry in the product A is equal to sum of the products of corresponding entries from the $i^th$ row of and from all the entries of $x$
Example. \[A = \left[ {\begin{array}{*{20}{c}}1&2&1\\3&2&4\\1&1&3\end{array}} \right],\,\,{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\]Their Product: \[\begin{array}{l}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}1&2&1\\3&2&4\\1&1&3\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\\ = {x_1}\left[ {\begin{array}{*{20}{c}}1\\3\\1\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}2\\2\\1\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}1\\4\\3\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{{x_1} + 2{x_2} + {x_3}}\\{3{x_1} + 2{x_2} + 4{x_3}}\\{{x_1} + {x_2} + 3{x_3}}\end{array}} \right]\end{array}\]Thus, every entry in the product is sum of the products.
Therefore,
TRUE

Step 5: (e)
Statement: If the columns of an $m \times n$ matrix A span $R^m$, then the equation $A{\bf{x}} = b$ is consistent for each b in $R^m$.

The Statement is True. The statement is same as any of the statement given below:
(1) The matrix has a pivot element in each row.
(2) The vector b is a linear combination of columns of A for each b in $R^m$
(3) $Ax=b$ has a solution for each b in $R^m$
(4) The columns of A span $R^m$

Therefore,
TRUE

Step 6: (f)
Statement: If A is an $m \times n$ matrix and if the equation $Ax=b$ is inconsistent for some b in $R^m$, then A cannot have a pivot position in every row.

The Statement is True. The statement is same as any of the statement given below:
(1) The matrix has a pivot element in each row.
(2) The vector b is a linear combination of columns of A for each b in $R^m$
(3) Ax=b has a solution for each b in $R^m$
(4) The columns of A span $R^m$

Therefore,
TRUE