In each case, determine if the statement is TRUE or FALSE and give a reason for
ID: 3030690 • Letter: I
Question
In each case, determine if the statement is TRUE or FALSE and give a reason for your choice. (a) The solution set to the matrix equation Ax = b is the span of a certain set of vectors. (b) The columns of a 4 times 5 matrix form a linearly dependent set. (c) If the columns of the coefficient matrix A are linearly independent, then the equation Ax = b is always consistent. (d) If V_1, V_2, V_3 and V_4 are in R^4 and the set {V_1, V_2, V_3} is linearly dependent, then the set {v_1, V_2, V_3, V_4} is linearly dependent. (e) The transformation T with rule T([x_1 x_2]) = [2x_1 - 3x_2 x_1 + 4 5x_2] is a linear transformation. (f) If T: R^n rightarrow R^m is a linear transformation, then T(O) = O. (g) If an augmented matrix [A | b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.Explanation / Answer
The statement is FALSE because if A is not invertible, then Ax = b will have either no solution, or an infinite number of solutions. If it has an infinite number of solutions, then the solution set cannot be the span of a certain set of vectors. Similarly if the system Ax = b has no solution, the statement is FALSE. For the statement to be TRUE, it is necessary that it be true irrespective of the fact that A is invertible or not. The columns of a 4x5 matrix form a linearly dependent set. This is a TRUE statement as there will be at least one free variable in the general solution of Ax = 0 (since there are 5 variables and at most 4 pivots). Thus there are an infinite number of solutions, not just the trivial solution. If the columns of the coefficient matrix A are linearly independent, then the equation Ax = b is always consistent. TRUE as the matrix A is invertible when its columns are linearly independent. Then x = A-1b is a solution. In R4 if v1, v2, v3 are linearly dependent, then v1, v2 , v3 , v4 are also linearly dependent. TRUE. If v1, v2, v3 are linearly dependent, then there exist scalars a, b, c such that av1 + bv2 + cv3 = 0. Then av1 + bv2 + cv3 + o.v4 = 0.Thus, v1, v2 , v3 , v4 are linearly dependent. The statement is FALSE. If x = (x1 , x2 )T and y = ( y1 , y2 )T , Then T(x+y) T(x) + T(y) as x1 + 4 + y1 +4 (x1 + y1 +4). If T: Rn Rm is a linear transformation, then T(0) = 0. TRUE. A Linear transformation preserves the operations of vector addition and scalar multiplication. Hence, if v be any vector in the domain of T,then T(0) = T(0v) = 0T(v) = 0. If an augmented matrix [A | b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.FALSE, because any matrix can be reduced to reduced echelon form by a finite sequence of elementary column operations
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