midterm 2

Question 1

if vector b is in the span of v vectors

Answer 1

when in augmented form it is consistent

Question 2

if the span of vectors can produce any vector in Rm

Answer 2

there must be a pivot in every row

Question 3

if the span of vectors can not produce all vectors in Rm

Answer 3

there must not be a pivot in every row

Question 3

if the the span of vectors can produce all vectors in Rm in infinitely many ways

Answer 3

there must be a pivot in every row but not a pivot in every column

Question 3

if the vectors are linearly independent

Answer 3

the vectors have a pivot in every column

Question 4

If T(x) = Ax is onto

Answer 4

A has a pivot in every row

Question 4

If T(x) = Ax is 1-to-1

Answer 4

A has a pivot in every column (lin indep)

Question 4

A (nxn) is invertible

Answer 4

A can be transformed into Identity matrix

Question 5

Why would an mxn matrix where m>n be not onto

Answer 5

There is not a pivot in every row

Question 6

Why would an mxn matrix where m<n not be one to one

Answer 6

there are fewer pivots than columns

Question 7

If A is a square one to one and onto then

Answer 7

A is invertible

Question 8

If we transform from R2 to R5 and it is not onto

Answer 8

2 columns with 5 rows: fewer pivots than rows

Question 9

Why is it that if A is invertible it is onto

Answer 9

pivot in every row (A can be reduced to identity matrix)

Question 10

If the vectors in A are not lin indep, why is it not 1-to-1

Answer 10

if lin dep, there is not a pivot in every column

Question 11

If Ax = b always has a solution is it onto or one-to-one

Answer 11

onto (not always 1-to-1)

Question 12

If Ax = b always has inf many solutions is it onto or 1-to1

Answer 12

onto (never 1-to1)

Question 13

If Ax = b never has inf many solutions is it onto or 1-to1

Answer 13

1-to-1 for sure but not necessarily onto

Question 14

If Ax = b always has a unique solution is it onto or 1-to1

Answer 14

1-to-1 and onto