Linear Algebra Exam 2 - Terms and Concepts

If A = [aij] and B= [bij] are mxn matrices and C is within R, we have the following definitions:

(1) A = B if aij = bij (2) A + B = [aij + bij] (3) CA = [Caij]

Equivalent statements that make up the Invertible Matrix Theorem:

(1) A is an invertible matrix (2) A is row equivalent to the nxn identity matrix (3) A has n pivots (4) The equation Ax = 0 has only the trivial solution (5) The columns of A form a linearly independent set (6)The linear transformation x —> Ax is one-to-one (7) The equation Ax = b has at least one solution for each b in R^n (8) The columns of A span R^n (9) The linear transformation A —> Ax maps R^n onto R^n (10) There is an nxn matrix C such that CA = I (11) There is an nxn matrix D such that AD = I (12) A^T is an invertible matrix

Types of Contraction/Expansion Standard Matrices

(1) Horizontal Contraction and Expansion (2) Vertical Contraction and Expansion

Types of Shear Standard Matrices

(1) Horizontal Shear (2) Vertical Shear

Properties of the Inverse:

(1) If A is an invertible matrix, then A^-1 is invertible and (A^-1)^-1 = A (2) If A and B are mxn invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. I.e. (AB)^-1 = B^-1A^-1 (3) If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^-1. I.e. (A^T)^-1 = (A^-1)^T

Types of Projection Standard Matrices

(1) Projection onto the x1-axis (2) Projection onto the x2 axis

Types of Reflection Transformation Matrices

(1) Reflection through the x1 axis (2) Reflection through the x2 axis (3) Reflection through the line x2 = x1 (4) Reflection through the line x2 = -x1 (5) Reflection through the origin

Two definitions of matrix multiplication:

(1) The column definition (2) The hand-twist method

Let a be an mxn matrix. If A is invertible, AX = b:

(1) has a solution (2) the solution is unique

Properties of matrix multiplication: A(BC) =

(AB)C. Matrix multiplication is always associative

Properties of matrix multiplication: r(AB) = ___ = ____, if r is in R

(rA)B; A(rB)

Concerning matrices: r(sA) =

(rs)A

Concerning matrices: A + 0 = _____ = _____

0 + A; A

A transformations a contraction if k is...

0<k<1

THEOREM 3: Let A and B denote matrices whose sizes are appropriate for the following sums and products: A(^T)^T =

A

Concerning matrices: (A + B) + C =

A + (B + C)

Determinant

A function that inputs a matrix nxn and outputs a real number. It's closely tied to the question of whether a matrix is invertible

One-to-One

A mapping T: R^n —> R^m in which each b in R^m is the image of AT MOST ONE x in R^m

Properties of matrix multiplication: A(B + C) =

AB + AC. Matrix multiplication is always distributive

Properties of the inverse: If A and B are mxn invertible matrices, then so is ____, and the inverse of AB is the product of the inverses of A and B in the ___ order

AB; reverse (i.e. (AB)^1 = B^-1A^-1)

Properties of matrix multiplication: If In is the nxn identity matrix, then InA = ___ = _____

AIn; A

Equation for the Inverse of a Two-By-Two Matrix

A^-1 = (1/(ad - bc)[(d,-b),(-c,a)])

Let A and B be square matrices. If AB = I, then A and B are both invertible, with B = _____ and A = _____

A^-1; B^-1

The inverse of A, if it exists, is a matrix A^-1 such that AA^-1 = ___ = ____

A^-1A; In

THEOREM 3: Let A and B denote matrices whose sizes are appropriate for the following sums and products: (A + B)^T =

A^T + B^T

Properties of the inverse: If A is an invertible matrix, then so is ___, and the inverse of A^T is the ______

A^T; transpose of A^-1 (i.e. (A^T)^-1 = (A^-1)^T)

If A is an mxn matrix, and if B is an nxn matrix with columns b1, ..., bp, then the produce AB is the mxn matrix whose columns are ______

Ab1, ..., Abp

Transpose

An mxn matrix A^T whose columns are the corresponding rows of the mxn matrix A

Elementary Matrix

An mxn matrix obtained by performing one elementary row operation to In. You can write any series of row operations as a sequence of elementary row operations!

Concerning matrices: A + B =

B + A

(AB)^T =

B^T A^T

Concerning matrices: C(A + B) =

CA + CB

(C + D)A =

CA + DA

If we want to know if a matrix is invertible, sometimes we can ______

Check for a dependency relation! If a dependency relation exists, the matrix is dependent and, thus, not invertible

Commutable

Describes A and B is AB = BA

Not singular

Describes a matrix that is invertible

Singular

Describes a matrix that is not invertible

If something is linear, it _____ and ______

Distributes over addition; you can pull constants out

If an elementary row operation is performed on an mxn matrix A, the resulting matrix can be written as ___, where the mxn matrix E is created by performing the same row operations to ____

EA; Im

A mapping T: R^n —> R^m is said to be onto R^m if...

Each b in R^m is the image of AT LEAST ONE x in R^n.

A transformation is a type of ______

Function

Transformation =

Function

Linear transformations are determined completely by what they do to the columns of ____

I2

Invertible Matrix Theorem

IMT. Series of equivalent terms that describes Matrix A (staring with "A is invertible")

THEOREM 7: An mxn matrix is invertible if and only if it is row equivalent to _____, and in this case, any sequence of elementary row operations that reduces A to In also ______

In ; transforms In into A^-1

Each elementary matrix E is _______

Invertible

All linear transformations from R^n to R^m are _______

Matrix transformations

Zero Matrix

Matrix whose entries are all zero

If AB = BC, is it true in general that B = C?

No

If a product AB is the zero matrix, can you conclude in general that either A = 0 or B = 0?

No!

Properties of matrix multiplication: Is AB = BA?

No! AB =/= BA. Matrix multiplication is not commutative in general

Do all laws of basic algebra apply to matrix operations?

No! For example, matrices AB = 0 does NOT imply that A = 0 or B = 0

Are projection transformations reversible?

Nope

Transformations are reversible only when they are ______

One-to-one

T is onto R^m when the ____ of T is ________.

Range; of the codomain R^m

The Algorithm Method for Finding A^-1

Row reduce the augmented matrix [AI]

A linear transformation T:R^n —> R^m is invertible if there's a linear transformation S:R^n —> R^n such that _______

S(T(x)) = T(S(x)) = X (we often write T^-1 for S)

How to find the inverse of a transformation matrix

STEP 1: Find A STEP 2: Find A^-1 STEP 3: Solve for T^-1 using A^-1

How to check if a transformation from R^n —> R^m is onto or one-to-one:

STEP 1: Reduce to echelon form STEP 2: Count the number of pivots STEP 3: If the number of pivots is equal to m, the transformation is onto. If the number of pivots is equal to n, then T is one-to-one

Diagonal Matrix

Square nxn matrix whose diagonal entries are zero

Basic Transformation Equation

T(x) = Ax

Let T:R^n—>R^m be a linear transformation. There exists a (unique) Matrix A such that _____

T(x) = Ax

In a one-to-one mapping, if x1 =/= x2, then...

T(x1) =/= T(x2)

A^-1

The inverse of A

Standard Matrix

The matrix A

a32 denotes ______

The third row and second column in Matrix A

Shear Transformation

Transformation that warps the shape of your vector

T is one-to-one if, for each b in R^m, the equation T(x) = b either has a _____ or _____

Unique solution; none at all

Are rotation transformations reversible?

Yep

Does the IMT apply ONLY to square matrices?

Yes

Transformation Matrix for Reflection Through the Origin

[(-1,0),(0,-1)]

Transformation Matrix for Reflection Through the X2-Axis

[(-1,0),(0,1)]

Transformation Matrix for Reflection Through the Line X2 = -X1

[(0,-1),(-1,0)]

Standard Matrix for Projection Onto the X1-Axis

[(0,0),(0,1)]

Transformation Matrix for Reflection Through the X1-Axis

[(1,0),(0,-1)]

Standard Matrix for Projection Onto the X1-Axis

[(1,0),(0,0)]

Vertical Shear Standard Matrix

[(1,0),(k,1)]

Horizontal Shear Standard Matrix

[(1,k),(0,1)]

Rotation Transformation Standard Matrix

[(cos θ, -sinθ),(sinθ, cosθ)]

Transformation Matrix for a Horizontal Contraction and Expansion

[(k,0),(0,1)]

We denote the entry in the ith row and jth column as ___

aij

THEOREM 9: Let T: R^n —> R^n be a linear transformation and let A be the standard matrix for T. The T is invertible if and only if A is ______

an invertible matrix

To construct A,let ______

e1 = [1 0 ... 0], ... , en = [0 0 ... 0 1], then A[T(e1),T(e2),T(e3)]

An nxn (square) matrix is diagonal if aij = 0 when _____

i =/= j

If we want to know if a matrix is invertible, it's easiest to first check ______

if A has n pivots

Properties of the inverse: If A is an invertible matrix, then A^-1 is ____ and (A^-1)^-1 = ____

invertible; A

If A is an nxn matrix and k is a positive integer, then A^k denotes the product of ______

k copies of A

A transformation is an expansion if k is...

k>1

An mxn entry has ___ rows and ____ columns

m;n

A matrix is non-singular if it's determinant is ____

not zero

THEOREM 3: Let A and B denote matrices whose sizes are appropriate for the following sums and products: For any scalar r, (rA)^T =

rA^T

The transpose of a product of matrices equals the product of their transposes in ________

reverse order

All elementary row operations are ______

reversible

If a matrix is not 2x2, you can find the inverse by ____________

setting your matrix equal to In and solving.

Each column of AB is a linear combination of the columns of A using weights from _______

the corresponding column of B

We can only add matrices together if _____. Otherwise, they are ________.

the same size; undefined

The inverse of matrix A, if it has one, is _____

unique

THEOREM 5: If A is an invertible nxn matrix, then for each b in R^n, the equation Ax = b has the unique solution _______

x = A^-1b

In a one-to-one mapping, if T(x1) = T(x2), then...

x1 = x2