Linear Algebra T/f Unit 2

Front 1

A linear transformation is a special type of function.

Back 1

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Front 2

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^3.

Back 2

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Front 3

If A is an m x n matrix, then the range of the transformation x --> Ax is R^m.

Back 3

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Front 4

Every linear transformation is a matrix transformation.

Back 4

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Front 5

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

Back 5

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Front 6

Every matrix transformation is a linear transformation.

Back 6

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Front 7

The codomain of the transformation x --> Ax is the set of all linear combinations of the columns of A.

Back 7

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Front 8

If T : R^n --> R^m is a linear transformation and if c is in R^m, then a uniqueness question is "Is c in the range of T?"

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Front 9

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

Back 9

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Front 10

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

Back 10

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Front 11

A linear transformation T : R^n --> R^m is completely determined by its effect on the columns of the n x n identity matrix.

Back 11

T

Front 12

If T : R^2 --> R^2 rotates vectors about the origin through an angle, then T is a linear transformation.

Back 12

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Front 13

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

Back 13

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Front 14

A mapping T : R^N --> R^m is onto R^m if every vector x in R^n maps onto some vector in R^m.

Back 14

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Front 15

If A is a 3 x 2 matrix, then the transformation x --> Ax cannot be one-to-one.

Back 15

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Front 16

Not every linear transformation from R^n to R^m is a matrix transformation.

Back 16

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Front 17

The columns of the standard matrix for a linear transformation from R^n to R^m are the images of the columns of the n x n identity matrix.

Back 17

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Front 18

The standard matrix of a linear transformation from R^2 to R^2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form [a 0, 0 d], where a and d are +-1.

Back 18

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Front 19

A mapping T : R^n --> R^m is one-to-one if each vector in R^n maps onto a unique vector in R^m.

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Front 20

If A is a 3 x 2 matrix, then the transformation x --> Ax cannnot map R^2 to R^3.

Back 20

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Front 21

If A and B are 2 x 2 with columns a1, a2, and b1, b2, respectively, then AB = [a1b1 a2b2].

Back 21

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Front 22

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

Back 22

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Front 23

AB + AC = A(B + C)

Back 23

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Front 24

A^T + B^T = (A + B)^T

Back 24

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Front 25

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

Back 25

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Front 26

If A and B are 3 x 3 and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3].

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Front 27

The second row of AB is the second row of A multiplied on the right by B.

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Front 28

(AB)C = (AC)B

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Front 29

(AB)^T = A^TB^T

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Front 30

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

Back 30

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Front 31

In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.

Back 31

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Front 32

If A and B are n x n and invertible, then A^-1B^-1 is the inverse of AB.

Back 32

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Front 33

If A = [a b, c d] and ab - cd != 0, then A is invertible.

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Front 34

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

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Front 35

Each elementary matrix is invertible.

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Front 36

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

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Front 37

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

Back 37

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Front 38

If A = [a b, c d] and ad = bc, then A is not invertible.

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Front 39

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

Back 39

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Front 40

If A is invertible, then elementary row operations that reduce A to the identity I, also reduce A^-1 to In.

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Front 41

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

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Front 42

If the columns of A span R^n, then the columns are linearly independent.

Back 42

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Front 43

If A is an n x n matrix, then the equation Ax = b has at least one solution for each b in R^n.

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Front 44

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

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Front 45

If A^T is not invertible , then A is not invertible.

Back 45

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Front 46

If there is an n x n matrix D such that AD = I, then there is also an n x n matrix C such that CA = I.

Back 46

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Front 47

If the columns of A are linearly independent, then the columns of A span R^n.

Back 47

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Front 48

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

Back 48

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Front 49

If the linear transformation x --> Ax maps R^n into R^n, then A has n pivot positions.

Back 49

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Front 50

If there is a b in R^n such that the equation Ax = b is inconsistent, then the transformation x --> Ax is not one-to-one.

Back 50

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Front 51

An n x n determinant is defined y the determinants of (n - 1) x (n - 1) submatrices.

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Front 52

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

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Front 53

The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row.

Back 53

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Front 54

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

Back 54

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Front 55

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

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Front 56

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

Back 56

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Front 57

If the columns of A are linearly dependent, then det A = 0.

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Front 58

det(A + B) = detA + detB.

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Front 59

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

Back 59

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Front 60

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

Back 60

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Front 61

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

Back 61

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Front 62

det A^T = (-1)detA.

Back 62

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