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GATECH / Math / MATH 1554 / Why is a linear transformation is a special type of function?

# Why is a linear transformation is a special type of function? Description

##### Description: Covers sections 1.1-3.3
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If you want to pass this class, use these notes. Period. I for sure will!

Midterm 1 Study Guide

## Why is a linear transformation is a special type of function?

Linear Algrebra | Spring 2016 | Chapters 1-3

Mark each statement true or false and justify answer.

1. Every elementary row operation is reversible.

2. A 5x6 matrix has six rows.

3. The solution set of a linear system involving variables x1,…,xn is a list of  numbers (s1,…,sn) that makes each equation in the system a true statement  when the values s1,…,sn are substituted for x1,…,xn, respectively.

4. Two fundamental questions about a linear system involve existence and  uniqueness.

5. Elementary row operations on an augmented matrix never change the solution  set of the associated linear system.

## How does two matrices are row equivalent if they have the same number of rows?

6. Two matrices are row equivalent if they have the same number of rows. 7. An inconsistent system has more than one solution.

8. Two linear systems are equivalent if they have he same solution set.

9. In some cases, a matrix may be row reduced to more than one matrix in  reduced echelon form, using different sequences of row operations.

10.The row reduction algorithm only applies to augmented matrices for a linear  system.

11.A basic variable in a linear system is a variable that corresponds to a pivot  column in the coefficient matrix.

12.Finding a parametric description of the solution set of a linear system is the  same as solving the system.

13.If one row in an echelon form of an augmented matrix is 0 0 0 5 0 ,  then the associated linear system is inconsistent.

## Whenever a system has free variables, the solution set contains many solutions, is what?

If you want to learn more check out How did the u.s. army change their tactics when fighting the natives?

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Midterm 1 Study Guide

Linear Algrebra | Spring 2016 | Chapters 1-3 We also discuss several other topics like Protection myth used by whom?

14.The echelon form of a matrix is unique.

15.The pivot positions in a matrix depend on whether row interchanges are used  in the row reduction process.

16.Whenever a system has free variables, the solution set contains many  solutions.

17.Another notation for the vector −43 is [-4 3].

18.The points in the plane corresponding to −25 and −52 lie on a line through  the origin.

19.An example of a linear combination of vectors v1 and v2 is the vector '(�'.

20.The solution set of the linear system whose augmented matrix is  �' �( �+ � is the same as the solution set of the equation  �'�' + �(�( + �+�+ = �. Don't forget about the age old question of What are the two types of weathering?

21.The set Span{u,v} is always visualized as a plane through the origin. 22.Any list of five real numbers is a vector in ℝ5.

23.The vector u results when a vector u-v is added to the vector v. 24.The weights c1,…,cp in a linear combination c1v1+…+ cpvp cannot all be zero.

25.When u and v are nonzero vectors, Span{u,v] contains the line through u and  the origin.

26.Asking whether the linear system corresponding to an augmented matrix  �' �( �+ � has a solution amounts to asking whether b is in  Span{a1,a2,a3}.

2 We also discuss several other topics like What is assonance and its example?

Midterm 1 Study Guide

Linear Algrebra | Spring 2016 | Chapters 1-3

27.The equation Ax=b is referred to as a vector equation.

28.A vector b is a linear combination of the columns of a matrix A if and only if  the equation Ax=b has at least one solution. Don't forget about the age old question of What does gender gap mean?

29.The equation Ax=b is consistent if the augmented matrix � � has a pivot  position in every row.

30.The first entry in the product Ax is a sum of products.

31.If the columns of an mxn matrix A span ℝm, then the equation Ax=b is  consistent for each b in ℝm.

32.If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in  ℝm, then A cannot have a pivot position in every row.

33.Every matrix equation Ax=b corresponds to a vector equation with the same  solution set.

34.Any linear combination of vectors can always be written in the form Ax for a  suitable matrix A and vector x

35.The solution set of a linear system whose augmented matrix is  �' �( �+ � is the same as the solution set of Ax=b, if � = �' �( �+ .

36.If the equation Ax=b is inconsistent, then b is not in the set spanned by the  columns of A.

37.If the augmented matrix � � has a pivot position in every row, then the  equation Ax=b is inconsistent.

38.If A is an mxn matrix whose columns do not span ℝm, then the equation Ax=b is inconsistent for some b in ℝm.

39.A homogeneous equation is always consistent.

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Linear Algrebra | Spring 2016 | Chapters 1-3

40.The equation Ax=0 gives an explicit description of its solution set.

41.The homogenous equation Ax=0 has the trivial solution if and only if the  equation has at least one free variable.

42.The equation � = � + �� describes a line through v parallel to p.

43.The solution set of Ax=b is the seet of all vectors of the form � = � + �8,  where vh is any solution of the equation Ax=0.

44.If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.

45.The equation � = �(� + �+�, with x2 and x3 free variables (and neither u or v a multiple of the other), describes a plane through the origin.

46.The equation Ax=b is homogenous is the zero vector is a solution

47. The effect of adding p to a vector is to move the vector in a direction parallel  to p.

48.The solution set of Ax=b is obtained by translating the solution set of Ax=0.

49.The columns of a matrix A are linearly independent if the equation Ax=0 had  the trivial solution.

50.If S is a linearly independent set, then each vector is a linear combination of  the other vectors in S.

51.The columns of a 4x5 matrix are linearly dependent

52.If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span{x,y}.

53.Two vectors are linearly dependent if and only if they lie on a line through the  origin.

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Linear Algrebra | Spring 2016 | Chapters 1-3

54.If a set contains fewer vectors than there are entries in the vectors, then the  set is linearly independent.

55.If x and y are linearly independent, and if z is the Span{x,y}, then {x,y,z} is  linearly dependent.

56.If a set in ℝ:is linearly dependent, then the set contains more vectors than  there are entries in each vector.

57.A linear transformation is a special type of function.

58.If A is a 3x5 matric and T is a transformation defined by � � = ��, then the  domain of T is ℝ+.

59.If A is an mxn matrix, the the range of the transformation � ⊢ �� is ℝ?. 60.Every linear transformation is a matrix transformation.

61.A transformation T is linear if ant only if � �'�' + �(�( = �'� �' + �(� �( for  all v1 and v2 in the domain of T and for all scalars c1 and c2.

62.Every matrix transformation is a linear transformation

63.The codomain of the transformation � ⊢ �� is the set of all linear combinations  of the columns of A.

64.If �: ℝ: → ℝ? is a linear transformation and if c is in ℝ?, then a uniqueness  question is “Is c in the range of T?”

65.A linear transformation preserved the operations of vector addition and scalar  multiplication

66.The superposition principle is a physical description of a linear transformation.

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Linear Algrebra | Spring 2016 | Chapters 1-3

67.A linear transformation If �: ℝ: → ℝ? is completely determined by its effect on  the columns of the nxn identity matrix.

68.If If �: ℝ( → ℝ( rotates vectors about the origin through an angle �, then T is a  linear transformation.

69.When two linear transformations are performed one after another, the  combines effect may not always be a linear transformation

70.A mapping If �: ℝ: → ℝ? is onto If ℝ? if every vector x in If ℝ: maps onto  come vector in If ℝ?.

71.If A is a 3x2 matrix, then the transformation � ⊢ �� cannot be one-to-one. 72.Not every linear transformation from ℝ: to ℝ? is a matrix transformation

73.The columns of the standard matrix for a linear transformation from ℝ: to ℝ? are the images of the columns of the nxn identity matrix.

74.The standard matrix of a linear transformation from ℝ( to ℝ( that reflects points  through the horizontal axis, the vertical axis, or the origin has the form � 0

0 � ,

where a and d are ± 1.

75.A mapping If �: ℝ: → ℝ? is one-to-one if each vector in If ℝ: maps onto a  unique vector in If ℝ?.

76.If A is a 3x2 matrix, then the transformation � ⊢ �� cannot map ℝ( onto ℝ+.

77.If A and B are 2x2 with columns a1,a2 and b1,b2, respectively, then  �� = �'�' �(�( .

78.Each column of AB if a linear combination of the columns of B using weights  from the corresponding column of A.

79.�� + �� = �(� + �)

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Linear Algrebra | Spring 2016 | Chapters 1-3

80.�K + �K = (� + �)K

81.The transpose of a product of matrices equals the product of their transposes  in the same order.

82.If A and B are 3x3 and � = �' �( �+ , then �� = ��' ��( ��+ . 83.The second row of AB is the second row of A multiplied on the right by B. 84. �� � = �� �

85.(��)K = �K�K

86.The transpose of a sum of matrices equals the sum of their transposes

87.In order for a matrix B to be the inverse of A, both equation �� = � and  �� = � must be true.

88.If A and B are nxn and invertible, then �M'�M'if the inverse of AB. 89.If � = � �

� � and �� − �� ≠ 0, then A is invertible.

90.If A is an invertible nxn matric, then the equation Ax=b is consistent for each b in ℝ:.

91.Each elementary matrix is invertible.

92.A product of invertible nxn matrices is invertible, and the inverse of the product  is the product of their inverses in the same order.

93.If A is invertible, then the inverse of A-1 is A itself.

94.If � = � �

� � and �� = ��, then A is not invertible.

95.If A can be row reduced to the identity matrix, then A must be invertible.

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Midterm 1 Study Guide

Linear Algrebra | Spring 2016 | Chapters 1-3

96.If A is invertible, then elementary row operation that reduce A to the identity In also reduce A-1 to In.

97.If the equation Ax=0 has only the trivial solution, then A is row equivalent to  the nxn identity matrix.

98.If the columns of A span ℝ:, then the columns ate linearly independent.

99.If A is an nxn matrix, then the equation Ax=b has at least one solution for each  b in ℝ:.

100. If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.

101. If AT is not invertible, then A is not invertible.

102. If there is an nxn matrix D such that �� = �, there there is also an nxn  matric C such that �� = �.

103. If the columns of A are linearly independent, then the columns of A span ℝ:.

104. If the equation Ax=b has at least one solution for each b in ℝ:, then the  solution is unique for each b.

105. If the linear transformation � ⊢ �� maps ℝ: into ℝ:, then A has n pivot  positions.

106. If there is a b in ℝ: such that the equation Ax=b is inconsistent, the the  transformation � ⊢ �� is not one-to-one.

107. A subspace of ℝ: is any set H such that (i) the zero vector is in H, (ii)  u, v, and u+v are in H, and (iii) c is a scalar and cu is in H.

108. If v1,…,vp are in ℝ:, then Span{ v1,…,vp} is the same as the column  space of the matrix �' … �Q

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109. The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of ℝ?.

110. The columns of an invertible nxm matrix form a basis for ℝ:

111. Row operations do not affect linear dependence relations among the  columns of a matrix

112. A subset of H of ℝ: is a subspace if the zero vector is in H.

113. Given vectors v1,…,vp in ℝ:, the set of all linear combinations of these  vectors is a subspace of of ℝ:.

114. The null space of an mxn matrix is a subspace of of ℝ:. 115. The column space of a matrix A is the set of solutions of Ax=b.

116. If B is an echelon form of a matrix A, then the pivot columns of B forma  basis for Col A.

117. If � = {�', … , �Q} is a basis for a subspace H and if � = �'�' + ⋯ + �Q�Q,  then �', … , �Q are the coordinates of x relative to the basis �.

118. Each line in ℝ: is a one-dimensional subspace of ℝ:. 119. The dimension of Col A is the number of pivot columns of A.

120. The dimensions of Col A and Nul A add up to the number of columns  of A.

121. If a set of p vectors spans a p-dimensional subspace H of ℝ:, then  these vectors form a basis for H.

122. If � is a basis for a subspace H, then each vector in H can be written in  only one way as a linear combination of the vectors in �.

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123. If � = {�', … , �Q} is a basis for a subspace H of ℝ:, then the  correspondence � ⊢ [�]Y makes H look and act the same as ℝQ.

124. The dimension of Nul A is the number of variables in the equation  Ax=0.

125. The dimension of the column space of A is rank A.

126. If H is a p-dimensional subspace of ℝ:, then a linearly independent set  of p vectors in H is a basis for H.

127. An nxn determinant is defined by determinant of � − 1 × (� − 1) submatrices.

128. The (i,j)-cofactor of a matrix A is the matrix Aij obtained by deleting  from A its ith row and jth column.

129. The cofactor expansion of det A down a column is equal to the  cofactor expansion along a row.

130. The determinant of a triangular matrix is the sum of the entries on the  main diagonal.

131. A row replacement operation does not affect the determinant of a  matrix.

132. The determinant of A is the products of the pivots in any echelon form  U of A, multiplied by (-1)r, where r is the number of row interchanged made  during row reduction from A to U.

133. If the columns of A are linearly dependent, then det A = 0. 134. det � + � = det� + det�

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Linear Algrebra | Spring 2016 | Chapters 1-3

135. If three row interchanges are made in succession, then the new  determinant equals the old determinant.

136. The determinant of A is the product of the diagonal entries in A.

137. If detA is zero, then two rows or two columns are the same, or a row or  column is zero.

138. det�M' = −1 det�

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