# MTSol - SOLUTIONS TO PRACTICE MIDTERM VERSION 2(1[20 True...

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Unformatted text preview: SOLUTIONS TO PRACTICE MIDTERM VERSION 2 (1) [20] True or false. Just write the word TRUE/T or FALSE/F . No expla- nation is necessary. (a) Let x 1 = (1 , , 1) T ,x 2 = (1 , 1 , 3) T , and x 3 = (2 , 1 , 4) T . The set { x 1 ,x 2 ,x 3 } spans R 3 . FALSE (b) Let A,B ∈ R n × n such that AB = 0. Then rank( A ) + rank( B ) ≤ n . TRUE (c) Let f 1 ( x ) = e ax and f 2 ( x ) = e bx . f 1 and f 2 are linearly independent if a 6 = b . TRUE (d) If the set of vectors { u 1 ,...,u n } is a basis for a vector space V , then u 1 ,ldots,u n ,u n +1 are linearly dependent. TRUE (e) If dim( V ) = n , then there exists a set of n + 1 vectors in V that spans V . TRUE (f) Let A ∈ R m × n . If dim( N ( A )) = 0, the column vectors of A are linearly independent. TRUE (g) The set of all vectors of the form ( a,a + b,a + b + c + 3) is a subspace of R 3 . TRUE (h) If dim( V ) = n , then any set of n- 1 vectors in V must be linearly independent. FALSE (i) If A ∈ R m × n , rank( A ) ≤ min( m,n ). TRUE (j) If a set S of vectors in V contains the zero vector, then S is linearly dependent. TRUE (2) [15] What are the maximum and minimum possible ranks of a 5 × 3 matrix....
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MTSol - SOLUTIONS TO PRACTICE MIDTERM VERSION 2(1[20 True...

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