In order to determine if the vector w = (2, -6, 3) is a linear combination of th

Question

In order to determine if the vector w = (2, -6, 3) is a linear combination of the vectors u = (1, -2, -1) and v = (3, -5, 4) an equation au + bv = w was written to solve for constants ‘a’ and ‘b’. An echelon form of the augmented coefficient matrix led to no solution. Which of the following can be concluded from the above information?

1) w is not a linear combination of u and v.

2) u, v and w are linearly independent.

3) v is not a linear combination of u and w.

Statement (1) alone is true

Statement (2) alone is true

Statement (3) alone is true

Statements (1) and (2) are true

Statements (1) and (3) are true

Statements (2) and (3) are true

All the statements are true

All the statements are false

A.

Statement (1) alone is true

B.

Statement (2) alone is true

C.

Statement (3) alone is true

D.

Statements (1) and (2) are true

E.

Statements (1) and (3) are true

F.

Statements (2) and (3) are true

G.

All the statements are true

H.

All the statements are false

Explanation / Answer

option D) Statement 1 and 2 are true.

If there is no solution for augmented matrix, then there is no relationship between the given vectors w, u , v.

=> w, u , v are linearly independent.

=> w is not a linear combination of u and v.

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