Linear Algebra 1st Test T/F
A mapping T: ℝn maps to ℝm is one-to-one if each vector in ℝn maps onto a unique vector in ℝm.
False. A mapping T is said to be one-to-one if each b in ℝm is the image of at most one x in ℝn.
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.
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 matrix A: 3 vectors, each with 2 entries in each vector, this is considered 1 to 1. Is the linear transformation onto?
False. T is not one-to-one because the columns of the standard matrix A are linearly dependent. A is onto because the standard matrix span ℝ2 (pivot in every rows)
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 is a 3 x 5 matrix and T is a transformation defined by T(x) =Ax, then the domain of T is R cube (ℝ3).
False. The domain is actually ℝ5, because in the product Ax, if A is an m x n matrix then x must be a vector ∈ ℝn.
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.
If a set in ℝn is linearly dependent, then the set contains more vectors than there are entries in each vector.
False. There exists a set in set of real numbers ℝn that is linearly dependent and contains n vectors. One example is a set in set of real numbers ℝ2 consisting of two vectors where one of the vectors is a scalar multiple of the other.
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 columns of a matrix A are linearly independent if and only if Ax = 0 has no free variables, meaning every variable is a basic variable, that is, if and only if every column of A is a pivot column.
True
The columns of the matrix do not form a linearly independent set because the set contains more vectors than there are entries in each vector.
True
The set is linearly independent because neither vector is a multiple of the other vector.
True
The vector equation has only the trivial solution, so the vectors are linearly independent.
True
Given a reduced echelon matrix in ℝ3 is 1--1 iff the homogeneous system Ax = 0 has only the trivial solution iff there are no free variables.
True.
A linear transformation is a special type of function.
True. A linear transformation is a function from ℝn to ℝm that assigns to each vector x ∈ ℝn a vector T(x) ∈ ℝm.
Every matrix transformation is a linear transformation.
True. Every matrix transformation has the properties T(u + v) = T(u)+ T(v) and T(cu) = cT(u) for all u and v, in the domain of T and all scalars c.
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 an m x n matrix, then the columns of A are linearly independent if and only if A has n pivot columns. (square matrix)
True. If n is the number of columns, then pivot = columns will satisfy the statement
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
If x and y are linearly independent, and if z is in Span {x, y}, then {x, y, z} is linearly dependent
True. Since z is in Span{x, y}, z is a linear combination of x and y. Since z is a linear combination of x and y, the set {x, y, z} is linearly dependent.
The columns of the standard matrix for a linear transformation from ℝn to ℝm are the images of the columns of the n x n identity matrix under T
True. The standard matrix is the m x n matrix whose jth column is the vector T(ej), where ej is the jth column of the identity matrix in ℝn.
Every linear transformation from ℝn to ℝm is a matrix transformation.
True. There exists a unique matrix A such that T(x) = Ax for all x in ℝn.
A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2.
True. This equation correctly summarizes the properties necessary for a transformation to be linear.
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.
The range of the transformation x maps to Ax is the set of all linear combinations of the columns of A.
True; each image T(x) is of the form Ax. Thus, the range is the set of all linear combinations of the columns of A.
A linear transformation T: ℝn maps to ℝm always maps the origin of ℝn to the origin of ℝm.
True; for a linear transformation, T(0) is equal to 0.
