- Decide Whether The Following Statement Is True Or False If True Give A Short Explanation If False Provide An Example 1 (140.17 KiB) Viewed 21 times
Decide whether the following statement is TRUE or FALSE. If TRUE, give a short explanation. If FALSE, provide an example where it does not hold. (a) If Ağ = ő is consistent, then the set {6, ã1, ãa, ān} must be linearly dependent. Here a1, 2, ..., ān are the columns of A. (b) Any 5 vectors in R4 must be linearly dependent. (c) Any 3 vectors in R4 must be linearly independent. (d) The columns of any 4 x 5 matrix are linearly dependent. (e) Let {ū1, ū2, ū3} be a linearly independent set in R3, then the set {cũi, cūz, cū3} is linearly independent, for all scalars c+0. (f) There exists a linear transformation T:R3 → R such that - (O) -- - () T = 3 and T = 9. - = (8) There exist 3 linearly independent vectors ū1, ū2, ū3 satisfying the equation 2ū1 + 3ū2 – 4īz = 0. (h) If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. (i) If A is a 3 x 2 matrix with two pivot positions, then the equation Ağ= 7 has a nontrivial solution. (j) If A is a 3 x 3 matrix with two pivot positions, then the equation Ağ = b is consistent for every possible b.