Problem 1

Let , , , , be vectors in , as pictured

(a). Write in terms of , , (and not )

(b). Write in terms of , , (and not )

(c). Describe All the Vectors in that Can Be Written in terms of , , . then Describe All the Vectors in that Can Be Written in terms of Two of , ,

Any vector in can be written in terms of , , and because none of them are parallel, meaning we can also represent any vector in as a combination of just two of them.

Problem 2

Let , , and , , .

(a). Is Every Vector in also in ? either Prove that it Is, or Find a Vector Demonstrating it is not

Yes every vector in is also in as , , and are just linear combinations of , and .

(b). Is Every Vector in also in ? either Prove that it Is, or Find a Vector Demonstrating it is not

Every vector in is not also in as is not a linear combination of , , and .

(c). Is Every Vector in also in ? either Prove that it Is, or Find a Vector Demonstrating it is not

Because is a subset of as the , , and do not change the span.

does not span as there is no combination of , , and such that they sum to .

Problem 3

For all four parts of this question, let , , , and be vectors such that and .

(a). Is Every Vector in also in ? either Prove that it Is, or Find a Vector Demonstrating it is not

Yes every vector in is also in as and are just linear combinations of and .

(b). Is Every Vector in also in ? either Prove that it Is, or Find a Vector Demonstrating it is not

Yes every vector in is also in as and are just linear combinations of and .

(c). Suppose , , , and Are Vectors in . is is Necessarily True that Every Vector in is also in ?

No not every vector in is in as , , , and could all be identical parallel vectors.

(b). Suppose , , , and Are Vectors in . is is Necessarily True that Every Vector in is also in ?

No not every vector in is in as , , , and could all be identical parallel vectors.

Problem 4

In this question you will sketch a proof that a matrix has a unique reduced row echelon form. Let be a matrix. Suppose does not have a unique reduced row echelon form: let A be row equivalent to two different reduced row echelon form matrices, and .

Since and are different, there must be a “first different column” as you read the matrices from left to right. For each matrix, select that column as well as all the columns to the left that contain a pivot. Then delete all the other columns. This yields two new matrices, and , constructed from the columns of and , respectively, that differ only in their last column. Suppose

What might look like? Describe the possibilities, and by interpreting and as augmented matrices describing systems of equations, explain why all possibilities lead to contradictions.

has a unique solution at . If and differ only in their last column,

With no pivot we get

where , and because and must represent the same system of equations we get a contradiction as the only solution has to be at .

With a pivot we get

Which gives us an inconsistent linear system as .

Via the two contradictions on both cases we can conclude that can not be a unique matrix and must have only one reduced matrix.