Answer the following statements with TRUE or FALSE. If you answer TRUE, give a b

Question

Answer the following statements with TRUE or FALSE. If you answer TRUE, give a brief justification. If you answer FALSE, give a specific reason.

2.) Answer the following statements with TRUE or FALSE. If you answer TRUE, give a brišef justification. If you answer FALSE, give a specifie reason. (4pts each) (a.) If H is a vector subepace of a vector space V, then R is a linearly independent set. (b.) If one vector in a set of vectors is a scalar oultiple of another vector, then the set is linearly dependent. (e.) If a set of vectors lis linearly dependent, them one vector in the srt is a scnlar nultiple of another vector

Explanation / Answer

a) True

Let H be a subspace of a vector space V. An indexed set of vectors b1,…,bp in V is a basis for H if (i) is a linearly independent set.

b) True

If v,wv,w are linearly dependent so there is constants c1,c2c1,c2 which both are not zero at the same time and c1v+c2w=0c1v+c2w=0. Let c10c10 so by dividing, we have v=c2c1wv=c2c1w.

c) True

Linear Combination of Vectors

If one vector is equal to the sum of scalar multiples of other vectors, it is said to be a linear combination of the other vectors.

For example, suppose a = 2b + 3c, as shown below.

Note that 2b is a scalar multiple and 3c is a scalar multiple. Thus, a is a linear combination of b and c.

If v,wv,w are linearly dependent so there is constants c1,c2c1,c2 which both are not zero at the same time and c1v+c2w=0c1v+c2w=0. Let c10c10 so by dividing, we have v=c2c1wv=c2c1w.

c) True

Linear Combination of Vectors

If one vector is equal to the sum of scalar multiples of other vectors, it is said to be a linear combination of the other vectors.

For example, suppose a = 2b + 3c, as shown below.

11 16    =    2 1 2    +    3 3 4    =    2*1 + 3*3 2*2 + 3*4 a b c 2b + 3c

Note that 2b is a scalar multiple and 3c is a scalar multiple. Thus, a is a linear combination of b and c.

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