Problem 1

Let and . Is in the subset of spanned by the columns of ? Why or why not?

No is not as there is no combination of the columns of which produces .

Problem 2

(a). Do the Columns of Span ? Does the Equation Have a Solution for Each in ?

No neither of the statements in the question are true as there are less than 4 pivot points.

Problem 3

Let , , and . Does span ? Why or why not?

Yes it does as there are 3 pivot positions meaning they span

Problem 4

Could a set of three vectors in span all of ? Explain. What about vectors in when is less than ?

No a set of vectors with can never span as there will never not be at least one column which is linearly dependent.

Problem 5

Suppose is a matrix and is a vector in with the property that has a unique solution. Explain why the columns of must span .

For to only have one unique solution there must be 3 pivot points in the row echelon form of so the columns of must span .