Math 3A

Onto (Surjective)

#Pivots = #Rows

Parametric Vector Equation

(.3) x2(1) (0)

Echelon Form

1. All Zero Rows are at the Bottom 2. Every Entry below the Pivot Point is zero 3. All Pivot Points are to the right of the Previous Column.

Reduced Echelon Form

1. Be in Echelon Form First 2. Every entry above and below the pivot point is zero. 3. Every pivot point is zero

A system of linear equations has

1. No Solution 2. Exactly One Solution 3. Infinitely Many Solutions

Linearly Independent Rows

1. Pivot in each row 2. Onto (surjective) 3 .Ax = b has a solution for all b.

Linearly Independent Columns

1.Pivot in each column(After row reducing) 2.One to One 3. Ax=0 has ONLY x=0 as the solution

Homogenous

A system of linear equations that can be written as Ax = 0

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

FALSE - The trivial solution is always a solution to the equation Ax = 0.

I The codomain of the transformation x 7→ Ax is the set of all linear combinations of the columns of A.

FALSE The If A is m × n codomain is R m. The original statement in describing the range

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

FALSE. At least one entry in x is nonzero

TOF: Every linear transformation is a matrix transformation

False Every matrix transfromation is a linear transformation. There exists linear transformations that are not matrix transformations

TOF: If A is an mxn matrix, then the range of the transformation x to Ax in R^m

False The range of T is the set of linear combinations of the columns of A

TOF: If T: R^n to 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?"

False The uniqueness question is "Is c the image of a unique x in R^n"

TOF: If A is a 3x5 matrix and T is a Transformation defined by T (x) = Ax, then the domain of T is R^3

False. The domain of T is R^n when A has n columns.

Matrices Basic Operation

Interchange (Switch two rows) Scaling (Multiply a row by a constant) Replacement (Replace one row by the sum of itself and a multiple of another row)

Inconsistent

No Solution

Nontrivial Solution

Nonzero Vector that satisfies Ax=0

Consistent

One or Infinitely many solutions

Unique Solution

Only one solution

Linear Transformations

T: Linear Transformation U & V: vectors C: Scalar then the following holds: 1. T(U + V) = T(U) + T(V) 2. T(CV) = CT(V)

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

TRUE. If the zero vector is a solution then b = Ax = A0 = 0. So the equation is Ax = 0, thus homogenous.

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

True

TOF: 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 A transformation, T, is linear if T(u+v) = Tu + Tv for all u, v in domain T.

TOF: Every matrix transfromation is a linear transformation

True Every matrix transformation is a linear transformation

TOF: A linear transformation T: R^n to R^m always maps the origin of R^n to the origin of R^m

True If T is a linear transformation, T(0) = 0

TOF: The range of the transformation x to Ax is the set of all linear combinations of the columns of A.

True The range of T is the set of linear combinations of the columns of A because each imahe Tx is of the form AX

TOF: A linear transformation is a special type of function

True. The correspondance from x to Ax is a function from one set of vectors to another.

Trivial Solution

Zero Solution (x=0)

Linearly Independent

{v1....vp} in R^n is LI if the vector equation x1v1 + x2v2 +...+ xpvp = 0 has only a trivial solution

Linearly Dependent

{v1....vp} is LD if c1v1 +c2v2 + ...+ cpvp = 0 where not all the C's are not zero.