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Quiz 04, Math 205B: Conditions for Vector in Span of Column Vectors, Exercises of Linear Algebra

West Bengal State UniversityLinear Algebra

A math quiz from a university course, specifically math 205b. It includes instructions for finding necessary and sufficient conditions for a vector b to be in the span of the column vectors a, verifying conditions, finding values of m and n for a vector d, determining if column vectors are linear combinations of each other, and checking linear independence.

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

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Math 205B Quiz 04 page 1 02/05/2010 Name
1. Let A=
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. Label the column vectors of Aas c1, . . . , c5.
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has RREF
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1A. Use the FACT to find any/all conditions on b1, . . . , b5which are necessary and sufficient for bto be in the span of
the column vectors A.
1B. Verify that c5does indeed satisfy the conditions in 1A.
1C. Find all values of mand nfor which dis in the span of the column vectors A.
1D. Is c5a linear combination of the first four column vectors of A? If so, find an explicit LC; if not explain why not.
1E. Is c1a linear combination of the last four column vectors of A? If so, find an explicit LC; if not explain why not.
1F. Is the set S={c1,...,c5}linearly independent? Explain using the definition or any theorem weā€™ve discussed.

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Math 205B Quiz 04 page 1 02/05/2010 Name

  1. Let A =

, b =

b 1

b 2

b 3

b 4

b 5

, and d =

m

n

3

2

1

. Label the column vectors of A as c 1 ,... , c 5.

FACT:

has RREF

1A. Use the FACT to find any/all conditions on b 1 ,... , b 5 which are necessary and sufficient for b to be in the span of

the column vectors A.

1B. Verify that c 5 does indeed satisfy the conditions in 1A.

1C. Find all values of m and n for which d is in the span of the column vectors A.

1D. Is c 5 a linear combination of the first four column vectors of A? If so, find an explicit LC; if not explain why not.

1E. Is c 1 a linear combination of the last four column vectors of A? If so, find an explicit LC; if not explain why not.

1F. Is the set S = {c 1 ,... , c 5 } linearly independent? Explain using the definition or any theorem weā€™ve discussed.