Linear Algebra 1st Test T/F
Two vectors are linearly dependent if and only if they lie on a line through the origin : True. Two vectors are linearly dependent if one of the vectors is a multiple of the other. Two such vectors will lie on the same line through the origin.
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. : False. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector.
If vector {v1, v2, v3} are in ℝ3 and v3 is not a linear combination of v1 & v2, then {v1, v2, v3} is linearly independent. : False. Vector {v1, v2} could have been a linear combination of the others, thus the three vectors are linearly dependent.
The vector equation has only the trivial solution, so the vectors are linearly independent. : True
If A is an m x n matrix, then the range of the transformation x maps to Ax is set of real numbers ℝm. : False. The range of the transformation is the set of all linear combinations of the columns of A, because each image of the transformation is of the form Ax.
Every linear transformation is a matrix transformation. : False. A matrix transformation is a special linear transformation of the form x maps to Ax where A is a matrix.
Let T: ℝn maps to ℝm be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in ℝn. : True. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1 + c2v2 + c3v3 = 0. It follows that c1T(v1) + c2T(v2) + c3T(v3) =0. Therefore, the set T(v1), T(v2), T(v3) is linearly dependent.
If A is a 4 x 3 matrix, then the transformation x maps to Ax maps ℝ3 onto ℝ4. : False. The columns of A do not span ℝ4.
If a reduced echelon matrix T(x) = 0 has a row of [ 0 . . 0 | 0] or [0 . . .0 | b] , where b =/= 0, it's considered one to one. : False. If the matrix has the form of [ 0 . . 0 | 0] or [0 . . .0 | b] , where b =/= 0, then we would have a free variable, thus having a free variable will not be one to one since it's nontrivial solution
Given a reduced echelon matrix in ℝ3 onto iff for every vector b in ℝ3, Ax = b has a solution iff every row of A has a pivot. : True. Moreover, every row has a pivot, so the linear transformation T with standard matrix A maps R4 onto R3
