Determine whether the following statement is true or false a

Determine whether the following statement is true or false, and explain your answer shortly.

a) v=[3,2] and u=[-2,-3] are linearly independent.

b) If u is in Span {p,q}, then Span {p,q} = Span {p,q,u}

c) Let A be a 4x5 matrix. The number of pivots in the reduced echelon form of A must be either 4 or 5.

Solution

a) v=[3,2] and u=[-2,-3] are linearly independent.

This is True because there for linearly independence av+bu=0, means a and b must be 0

we we form equations then we get 3a-2b=0 and 2a-3b=0

whose solution is clearly a=b=0

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b) If u is in Span {p,q}, then Span {p,q} = Span {p,q,u}

Given that u is in Span {p,q} so u can be formed using p and q like u=ap+bq so Span {p,q} and Span {p,q,u}, represent same thing

Hence it is True.

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c) Let A be a 4x5 matrix. The number of pivots in the reduced echelon form of A must be either 4 or 5.

False as number of povots can be less too