True. 2 vectors are linearly dependence if among the vectors is a lot of of the other. Two such vectors will certainly lie ~ above the same line v the origin.

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 c1​T(v1​) + c2​T(v2​) + c3​T(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.

See more: Watch Naked And Afraid Xl Season 2 Episode 7 ): The Sickness Summary

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.
})}else;window.location.assign("https://brickandmortarphilly.com/explanations/textbook-solutions/linear-algebra-and-its-applications-5th-edition-9780321982384");">

})}else;window.location.assign("https://brickandmortarphilly.com/explanations/textbook-solutions/differential-equations-and-linear-algebra-4th-edition-9780134497181");">

})}else;window.location.assign("https://brickandmortarphilly.com/explanations/textbook-solutions/algebra-lineal-una-introduccion-moderna-spanish-edition-9786074816082");">

})}else;window.location.assign("https://brickandmortarphilly.com/explanations/textbook-solutions/elementary-linear-algebra-applications-version-10th-edition-9780470432051");">