Linear Algebra 1st Test T/F

Lakukan tugas rumah & ujian kamu dengan baik sekarang menggunakan Quizwiz!

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 c1​T(v1​) + c2​T(v2​) + c3​T(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.


Set pelajaran terkait

Chapter 16: Gastrointestinal and Urologic Emergencies

View Set

EF3 Upper Int 7A Frequently Confused Verbs

View Set

Definitions of flexion/extension/abduction ect.

View Set

Sedimentary Petrology Oppo Exam 2 (excluding diagenesis)

View Set

Three Fifths Compromise & Great Compromise

View Set

5. English Grammar Writing Conventions: Colons, Semicolons, and Periods

View Set

Chapter 14 Science cornell notes- The outer planets

View Set

Chapter 32: Assessment of Hematologic Function and Treatment Modalities

View Set

Alexander Fleming: Discovery of Penicillin

View Set