Suppose A is a 3 x3 matrix and b is a vector in R3 with the property what Ax- b has a unique solution. Explain why the columns of A must span R3 Choose the correct answer below.
a. The equation has a unique solution so for each pair of vectors x and b there is only one possible matrix A. Therefore the columns of A must span R3
b. The reduced echelon form of A must have a pivot in each row or there would be more than one possible solution for the equation Ax-b. Therefore the columns of A must span R3
c. Matrix A is a square matrix, so when computing Ax, the row-vector rule shows that the columns of A must span IR3
d. When b is written as a linear combination of the columns of A it simplifies to the vector of weights x Therefore the columns of A must span R3

A. 1=45 degress because 4 and 1 are vertical angles which means they are equal.

2= 135 degrees because angle 4+2 equals 180 degrees(straight line) so since we know that angle 4 is 45 degrees, you can subtract it from 180, which is 135.

3=135 degrees because it is vertical angles with angle 2 which is 135 degrees.

6= 45 degrees because we know angle 5 is 135 degress. So, 135+angle 6=180(straight line). So if you subtract 135 from 180, you get 45.

7=45 degrees because it is vertical angles with angle six which is 45 degrees.

8= 135 degrees because it is vertical angles with angle five which is 135 degrees.

10= 90 degrees, because we know that angle 10 and 11 are equal and that they equal 180 degrees together(straight line). So, if you divide 180 by two(two angles) you will get 90 degrees.

11= 90 degrees for the same reason as angle 10.

9=90 degrees because it is vertical angles with angle 11 which is 90 degrees.

12=90 degrees because it is vertical angles with angle 10 which is 90 degrees.

B.

1) 4 and 1 are vertical angles which makes them equal

2) 7 and 5 are supplementary angles, which means they add up to 180 degrees.

3) 9 and 10 are also supplementary angles, but the line dividing them is their angle bisector, splitting them into two equal parts, 90 degrees and 90 degrees.

Sorry this explaination is so long. I hope it's helpful!!