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.

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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 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 Spanx, 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.
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 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)
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.
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 the matrix do not form a linearly independent set because the set contains more vectors than there are entries in each vector.
True. A linear transformation is a function from ℝn to ℝm that assigns to each vector x ∈ ℝn a vector ​T(x​) ∈ ℝm.
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.
False. A matrix transformation is a special linear transformation of the form x maps to Ax where A is a matrix.
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.
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.
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 c1​T(v1​) + c2​T(v2​) + c3​T(v3​) =0. ​Therefore, the set ​T(v1​), ​T(v2​), ​T(v3) 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.
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 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.

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True. Moreover, every row has a pivot, so the linear transformation T with standard matrix A maps R4 onto R3
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.
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