Things to Know Midterm 1

Following are equivalent for mxn matrix A (consistency):

1. Ax = b consistent ∀b

2. Columns of A span the set R^m

3. A has a pivot in every row

4. b is a linear combo/in span of columns of A

How to determine if the columns of an mxn matrix A are linearly independent? What 2 things should you always check first?

Trivial solution is the only solution to Ax = 0

If n > m (more columns than rows), then automatically dependent; 0 vector always causes dependence

Following are equivalent for an mxn matrix A (independence):

1. Columns of A are linearly independent

2. Ax = 0 has only the trivial solution

3. A has a pivot in every column

4. A has no free variables

What is true about every consistent linear system in terms of solution sets and the homogenous system?

Every consistent linear system has a general solution of the form

x = x_p+x_h

x_p= particular solution (main solution of Ax = b)

x_h= homogenous solution (solution of Ax = 0)

What is a dependence relation? How do you form one?

Expression relating a set of dependent vectors

Multiply the free variable by (each entry of it multiplied by the corresponding entry of the matrix)

Example: x₃(5a₁+3a₂-2a₃) = 0

What are the rules for Echelon form? What about RREF?

EF:

1. 0 rows on bottom

2. Leading entries strictly behind those of above rows

3. All entries below a leading entry are 0

RREF (on top of EF):

1. Leading entries are 1s

2. All entries above leading entries are also 0

What does row equivalent mean?

You can obtain each matrix from the other with elementary row operations

What does vector equivalent mean? What’s a common trick question?

Every corresponding entry is the same

(1,2,3,0) ≠ (1,2,3) (different spaces)

What is the span of 1 vector? What about 2 non-colinear?

A line

A plane

What does non colinear mean?

Not on the same line

What’s the form of a linear map (function) and what are the inputs and outputs called? Actual outputs? What does image and pre-image mean?

T: Rⁿ → R^m

Domain → codomain

Actual outputs → range

Image (b) is was you get from putting the pre-image (x) through T (under T)

What are the two rules for a linear transformation?

1. T(u + v) = T(u) + T(v)

   1. T(cu) = cT(u)

What does it mean if a map can be written T(x) = Ax? What’s A called? How do you find this special matrix?

It’s a matrix transformation (it’s linear)

Standard matrix of T (if it exists then T is linear)

A = [T(e₁) T(e₂) … T(e_n)] where e_i is the ith standard basis vector

What is the rotation matrix? What does it do?

cos -sin

sin cos

Rotates vector counterclockwise by theta wrt the origin

What does it mean for a map to be onto (surjective)? How do you determine this?

Every output has at least 1 input (every b is image of some x in Rⁿ)

|range of T| = |codomain of T|

Ax = b has solution for all b

What does it mean for a map to be one to one? How do you determine this?

No 2 inputs have the same output

|range of T| <= |codomain of T|

Columns of A are linearly independent

What does the laplace interpolation theorem say?

Given n points, the interpolating equation through all n is an n-1 degree polynomial

p(t) = a₀ + a₁t + a₂t² + … + a_n-1t^n-1

What is a contraction?

Shrink or stretch

What is shear?

Tilts the side(s) of the shape without changing volume or area

What is a projection?

Flatten into a line