linear final

0.0(0)
learnLearn
examPractice Test
spaced repetitionSpaced Repetition
heart puzzleMatch
flashcardsFlashcards
Card Sorting

1/33

flashcard set

Earn XP

Description and Tags

adsfasdf

Study Analytics
Name
Mastery
Learn
Test
Matching
Spaced

No study sessions yet.

34 Terms

1
New cards
  • How to tell if a matrix is one-to-one?

  • It is one to one if it only has trivial solution / columns are linearly independent.

2
New cards
  • How to tell if a matrix is onto?

  • If there is a pivot in every row.

3
New cards
  • If given radians remember:

  • cos -sin
    sin cos

4
New cards
  • If given T(es) = (points)

  • then it is columns.

5
New cards
  • If given T(es) = (equations)

  • then those are rows.

6
New cards

If a matrix is a rectangle

then it is linearly dependent.

7
New cards

If a matrix has a zero vector

then linearly dependent

8
New cards

If a matrix has a column that is a multiple of another

then linearly dependent

9
New cards
  • How to find the inverse of a 2x2 matrix?

10
New cards

How to solve a linear system using an inverse?

  • Take inverse and multiply it with b. Ax = b  -> A-1b = x

11
New cards

What happens if a triangle of zeroes below a diagonal?

  • The product of the diagonal is the determinant

  • Can also reduce to form the zero triangle to get the determinant

12
New cards

Cramer’s Rule

  1. Find determinant

  2. Replace the x,y,z columns with the RHS.

  3. Find dx/d, dy/d, dz/d

13
New cards

Find determinants of matrices (up to 4 × 4) (3.1 and 3.2)

14
New cards

Show that a set of vectors is a basis for a vector space (4.3)

  • An indexed set of vectors B in V is a basis for H if

  • B is linearly independent (check)

  • H spans B (find if it is invertible. If det 0, then it is invertible)

15
New cards
  • To find basis for Nul A (Kernel)

  • Solve Ax = 0

  • Write solution in parametric vector form

  • The vectors are are the solution

16
New cards
  • Find a basis for Col A

  • Reduce to echelon form

  • Identify pivots

  • The corresponding column in A itself form the basis

17
New cards
  • To find a basis for Row A

  • Reduce to echelon form

  • The nonzero rows will be the basis.

18
New cards
  • What is rank and nullity? 

  • Rank = Number of vectors for basis of Col A

  • Nullity = Number of vectors for basis of Nul A

  • Rank Nullity Theorem = Rank + Nullity = # of columns in original A

19
New cards

Find the coordinates of a given vector v (in terms of a basis B) and its corresponding coordinate vector [v]B (4.4)

  • Given coordinate vector and basis:

  • take the coordinate vector, and distribute it to the basis

20
New cards

Find the coordinates of a given vector v (in terms of a basis B) and its corresponding coordinate vector [v]B (4.4)

  • Given coordinate vector and basis:

  • take the coordinate vector, and distribute it to the basis

21
New cards
  • Find the coordinates of a given vector v (in terms of a basis B) and its corresponding coordinate vector [v]B (4.4)

    Given basis and x

  • : Augment basis with x.

22
New cards

Find the coordinates of a given vector v (in terms of a basis B) and its corresponding coordinate vector [v]B (4.4)
Use coordinates to check that polynomials are linearly dependent

  • Turn the polynomials into columns

  • Augment with zero. 

23
New cards

Determine if a given vector is an eigenvector of a given matrix (5.1)

  • If given vectors, multiply matrix with eigenvector

  • If the answer is a multiple of the vector, then it is an evector.

24
New cards

What happens if zero is along diagonal as an eigenvalue?

  • Matrix is not invertible

25
New cards

Determine if a given number λ is an eigenvalue of a given matrix A (by analyzing A − λI) (5.1)

  • Subtract matrix from the eigenvalue

  • Augment with zero, and see if it is linearly independent. 

  • If dependent, then it is an eigenvalue

  • If asked for the corresponding eigenvectors

    • Find the weird form. That is the vector

26
New cards

Find the eigenvalues of a matrix (5.2).

  • Subtract lambda from the matrix

  • Find determinant and set equal to zero

  • Factor to get eigenvalues.

27
New cards

Determine whether a given matrix is invertible based on its eigenvalues (5.2)

  • A matrix A is invertible if and only if all of its eigenvalues are non-zero.

28
New cards

Diagonalize a matrix (up to 3 × 3) by finding a diagonal matrix D and a matrix P such that A = PDP^−1 (5.3). 

  • To diagonalize a given matrix

    • Subtract diagonal by eigenvalues.

    • Find determinant 

    • Factor determinant, those are your eigenvalues

    • D = eigenvalues in diagonal

    • P = Find 3 linearly independent eigenvectors

      • Subtract matrix from eigenvalues, augment with zero.

      • Find weird form, that is the P.


29
New cards

Find the length of a vector (6.1).

  • Find norm, sqrt(u^2 + v^2)

30
New cards

Normalize a vector:

  • Find unit vector

31
New cards

Determine if two vectors are orthogonal

  • Dot product, if zero, then orthogonal

32
New cards

Show that a given set of vectors is an orthogonal basis for a subspace of R^n (6.2)

Find if all are orthogonal, then check determinant and LI

33
New cards

Find the projection of a vector onto a subspace (6.3).

<p></p>
34
New cards

Find the distance between a vector and a subspace (6.3).

<p></p>