Linear Alegebra T/f Unit 1

Front 1

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

Back 1

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Front 2

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

Back 2

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Front 3

The columns of any 4x5 matrix are linearly dependent.

Back 3

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Front 4

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

Back 4

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Front 5

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

Back 5

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Front 6

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

Back 6

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Front 7

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

Back 7

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Front 8

If a set in R^n is linearly dependent, then the set contains more vectors than there are entries in vectors.

Back 8

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Front 9

A homogeneous equation is always consistent.

Back 9

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Front 10

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

Back 10

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Front 11

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

Back 11

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Front 12

The equation x = p + tv describes a line through v parallel to p.

Back 12

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Front 13

The solution set of Ax = b is the set of all vectors of the form w = p + v, where v is any solution of the equation Ax = 0.

Back 13

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Front 14

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

Back 14

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Front 15

The equation x = x2u + x3v, with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin.

Back 15

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Front 16

The equation Ax = b is homogeneous if the zero vector is a solution.

Back 16

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Front 17

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

Back 17

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Front 18

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

Back 18

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Front 19

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

Back 19

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Front 20

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.

Back 20

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Front 21

The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row.

Back 21

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Front 22

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

Back 22

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Front 23

If the columns of an m x n matrix A span R^m, then the equation Ax = b is consistent for each b in R^m.

Back 23

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Front 24

If A is an m x n matrix and if the equation Ax = b is inconsistent for some b in R^m, then A cannot have a pivot position in every row.

Back 24

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Front 25

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

Back 25

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Front 26

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

Back 26

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Front 27

The solution set of a linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax = b, if A = [a1 a2 a3].

Back 27

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Front 28

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

Back 28

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Front 29

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

Back 29

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Front 30

If A is an m x n matrix whose columns do not span R^m, then the equation Ax = b is inconsistent for some b in R^m.

Back 30

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Front 31

Another notation for the vector [-4] . [ 3] is [-4, 3].

Back 31

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Front 32

The points in the plane corresponding to [-2] and [-5] . [ 5] [2] . lie on a line through the origin.

Back 32

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Front 33

An example of a linear combination of vectors v1 and v2 is a vector 1/2v1.

Back 33

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Front 34

The solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b.

Back 34

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Front 35

The set Span{u,v} is always visualized as a plane through the origin.

Back 35

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Front 36

Any list of five real numbers is a vector in R^5

Back 36

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Front 37

The vector u results when a vector u - v is added to the vector v.

Back 37

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Front 38

The weights c1, . . . . , cp in a linear combination c1v1 + . . . + cpvp cannot all be zero.

Back 38

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Front 39

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

Back 39

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Front 40

Asking whether the linear system corresponding to an augmented matrix [ a1 a2 a3 b] has a solution amounts to asking whether b is in Span {a1, a2, a3}.

Back 40

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Front 41

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

Back 41

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Front 42

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

Back 42

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Front 43

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

Back 43

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Front 44

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

Back 44

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Front 45

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

Back 45

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Front 46

The echelon form of a matrix is unique.

Back 46

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Front 47

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

Back 47

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Front 48

Reducing a matrix to echelon form is called the forward phase of the row reduction process.

Back 48

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Front 49

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

Back 49

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Front 50

A general solution of a system is an explicit description of all solutions of the system.

Back 50

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Front 51

Every elementary row operation is reversible.

Back 51

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Front 52

A 5 x 6 matrix has six rows.

Back 52

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Front 53

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.

Back 53

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Front 54

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

Back 54

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Front 55

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

Back 55

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Front 56

Two matrices are row equivalent if they have the same number of rows.

Back 56

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Front 57

An inconsistent system has more than one solution.

Back 57

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Front 58

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

Back 58

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