You are watching: Two vectors are linearly dependent if and only if they lie on a line through the origin
If a set contains under vectors than there are entries in the vectors, then the collection is linearly independent.
False. There exists a set that has fewer vectors 보다 there room entries in the vectors the is linearly dependent. One instance is a set consisting of 2 vectors where one of the vectors is a scalar lot of of the various other vector.
If x and also y are linearly independent, and if z is in expectations x, y, then x, y, z is linearly dependent
True. Due to the fact that z is in Spanx, y, z is a linear mix of x and also y. Because z is a linear combination of x and y, the set x, y, z is linearly dependent.
If a collection in ℝn is linearly dependent, climate the set contains more vectors 보다 there space entries in every vector.
False. There exists a collection in collection of real numbers ℝn the is linearly dependent and contains n vectors. One instance is a collection in set of genuine numbers ℝ2 consisting of 2 vectors where one of the vectors is a scalar multiple of the other.
If A is an m x n matrix, then the columns the A are linearly independent if and also only if A has n pivot columns. (square matrix)
The columns that a procession A room linearly elevation if and also only if Ax = 0 has actually no free variables, an interpretation every change is a basic variable, that is, if and also only if every column of A is a pivot column.
If vector v1, v2, v3 room in ℝ3 and v3 is no a linear combination of v1 & v2, then v1, v2, v3 is linearly independent.
False. Vector v1, v2 could have to be a linear combination of the others, hence the three vectors space linearly dependent.
The columns the the matrix execute not type a linearly independent set because the collection contains an ext vectors 보다 there are entries in every 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 also T is a revolution defined by T(x) =Ax, then the domain of T is R cube (ℝ3).
False. The domain is actually ℝ5, since in the product Ax, if A is one m x n procession then x have to be a vector ∈ ℝn.
If A is one m x n matrix, then the range of the change x maps to Ax is set of genuine numbers ℝm.
False. The variety of the revolution is the set of all linear combinations that the columns of A, due to the fact that each picture of the change is the the kind Ax.
False. A matrix transformation is a special linear transformation of the form x maps to Ax wherein A is a matrix.
A transformation T is straight if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain the T and for all scalars c1 and also c2.
True. This equation properly summarizes the properties vital for a transformation to be linear.
The variety of the revolution x maps come Ax is the collection of all straight combinations of the columns that A.
True; each image T(x) is the the form Ax. Thus, the range is the collection of all direct combinations that the columns that A.
True. Every matrix revolution has the properties T(u + v) = T(u)+ T(v) and T(cu) = cT(u) for every u and v, in the domain of T and all scalars c.
Let T: ℝn maps to ℝm it is in a linear transformation, and let v1, v2, v3 it is in a linearly dependent set in ℝn.
True. Provided that the collection v1, v2, v3 is linearly dependent, over there exist c1, c2, c3, not all zero, such the c1v1 + c2v2 + c3v3 = 0. It complies with that c1T(v1) + c2T(v2) + c3T(v3) =0. Therefore, the set T(v1), T(v2), T(v3) is linearly dependent.
The columns the the standard matrix because that a linear change from ℝn come ℝm are the pictures of the columns the the n x n identity matrix under T
True. The conventional matrix is the m x n matrix whose jth pillar is the vector T(ej), wherein ej is the jth column of the identity matrix in ℝn.
Given procession A: 3 vectors, each v 2 entries in each vector, this is considered 1 come 1. Is the straight transformation onto?
False. T is not one-to-one because the columns that the standard matrix A room linearly dependent. A is onto because the conventional matrix span ℝ2 (pivot in every rows)
If a lessened echelon matrix T(x) = 0 has a heat of < 0 . . 0 | 0> or <0 . . .0 | b> , where b =/= 0, it"s considered one come one.
False. If the matrix has actually the kind of < 0 . . 0 | 0> or <0 . . .0 | b> , where b =/= 0, then we would have actually a totally free variable, thus having actually a totally free variable will certainly not be one to one because it"s nontrivial solution
Given a lessened echelon procession in ℝ3 top top iff because that every vector b in ℝ3, Ax = b has actually a equipment iff every heat of A has actually a pivot.
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True. Moreover, every row has a pivot, for this reason the linear change T with standard procession A maps R4 top top R3
Given a lessened echelon matrix in ℝ3 is 1--1 iff the homogeneous device Ax = 0 has only the trivial equipment iff there space no complimentary variables.