25. For each of the following, decide if the statement is always true or always false or sometimes true, sometimes false

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25 For Each Of The Following Decide If The Statement Is Always True Or Always False Or Sometimes True Sometimes False 1 (976.28 KiB) Viewed 29 times

25. For each of the following, decide if the statement is always true or always false or sometimes true, sometimes false when the given condition is true. (a) Given: The vectors v1, V2, V3, V4, V5, V6 span Rº. Statement: The vectors v1, V2, V3, V4, V5, V6 are linearly independent. always true always false sometimes true, sometimes false (b) Given: B is a 6 x 6 matrix with det(B) = 0. Statement: The equation BX = b has infinitely many solutions. always true always false sometimes true, sometimes false (c) Given: B is a 6 x 6 matrix, b is in R6, BX = 0 has infinitely many solutions. Statement: The equation BX = b has infinitely many solutions. always true always false sometimes true, sometimes false (d) Given: B is a 6 x 6 matrix and the columns of B are linearly independent. Statement: The equation BX = b has exactly one solution. always true always false sometimes true, sometimes false (e) Given: E is a 6 x 8 matrix, b is in R6, N(E), the nullspace of E is 2-dimensional. Statement: The equation EX = b has infinitely many solutions. always true always false sometimes true, sometimes false (f) Given: D is a 7 x 5 matrix, b is in R?, N(D), the nullspace of D is 1-dimensional. Statement: The equation DX = b has infinitely many solutions. always true always false sometimes true, sometimes false