Linear algebra question (span?)

[ckp]

How would i go about telling if vector b (4 row, 1 column) is in the span of the columns of matrix A(4 row, 5 column)?

im just not sure what is asking, i know it would be an easy question if i knew what they meant by this.

 

[Billy Bob]

Same question reformulated: does the system Ax=b have a solution.

 

[ckp]

so would it not have a solution because there are 4 eq and 5 unknowns? also how would i justify in terms of span that there is no solution? (as opposed to just saying because there are 4 eq and 5 unknowns)

 

[Billy Bob]

Do you know the actual matrix A and the vector b? Form the augmented matrix [A|b] and row reduce to see if the system is consistent or not.

 

[ckp]

Is this possible with A being 4rows x 5 columns and b 4 rows 1 column?

 

[Billy Bob]

Maybe yes:

1 0 0 0 0 | 1
0 1 2 0 0 | 1
0 0 0 1 0 | 1
0 0 0 0 1 | 1

Maybe not:

1 0 0 0 0 | 1
0 1 2 0 0 | 1
0 0 0 1 0 | 1
0 0 0 0 0 | 1

Depends on A and b

 

[ckp]

So, say it is consistent. Then what do I do? (rr took a long time and numbers are rather large)

 

[Billy Bob]

If the system Ax=b is consistent, then the answer is "yes, b is in the span of the columns of A."

 

[ckp]

now, what if it is asked if a sub n is in the span of A (A consists of {a sub 1,...,a sub n})

 

[Billy Bob]

now, what if it is asked if a sub n is in the span of A (A consists of {a sub 1,...,a sub n})

Are you asking this: "is the rightmost column of A in the span of the columns of A?"

??