Life of Fred: Linear Algebra

34 flashcards · Mathematics · by lakesare

1. In the context of linear algebra - what are terms in Ax = b?

   A - coefficient matrix x - column vector of x1, x2, x3 b - column vector of constant terms Amxn * Xnx1 = Bmx1 Here is a system of linear equations: Here is a coefficient matrix A: Here is a column vector of constant terms b:

2. What's augmented matrix?

   Matrix A|B obtained by appending the columns of two given matrices. Usually for the purpose of performing the same elementary row operations on each of the given matrices.

3. Give the example of lower triangular matrix.

   A square matrix is called lower triangular if all the entries above the main diagonal are zero.

4. Give the example of upper triangular matrix.

   Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.

5. What's the matrix that's both lower triangular and upper triangular called?

   Diagonal matrix.

6. What's alliteration?

   When words begin from the same letter - "Peter Piper picked a peck of pickled peppers". Alliteration is the conspicuous repetition of identical initial consonant sounds in successive or closely associated syllables within a group of words.As a method of linking words for effect, alliteration is also called head rhyme or initial rhyme. "Alliteration" is from the Latin word littera, meaning "letter of the alphabet".

7. What's an identity element for some operation ⊙ and some set A?

   Identity element is some element i contained in set A, so that: for every a∈A, i⊙a = a⊙i = a.

8. What's the identity element for set 'real numbers', and operation +?

   0. a + 0 = 0 + a = a. Identity element is some element i contained in set A, so that: for every a∈A, i⊙a = a⊙i = a.

9. What's the identity element for set 'matrices', and operation * (matrix multiplication)?

   Identity matrix. A * I = I * A = A. Identity element is some element i contained in set A, so that: for every a∈A, i⊙a = a⊙i = a.

10. What linear system of equations is called homogeneous?

    System of equations of the form where all the constant terms are zero: Ax = 0.

11. What are 2 other ways to call NOT invertible matrix?

    degenerate singular A square matrix that is not invertible is called singular or degenerate.A square matrix is singular iff its determinant is zero.

12. Given the statement p→q, what are: contrapositive inverse converse

13. What variables are free here?

    x5 and x6 are free variables, because none of the equations start with x5 or x6. In reduced row echelon form we have four non-zero equations and six unknowns, therefore there are 6 − 4 = 2 free variables.

14. Give the solution to this system of equations: traditionally written solution as x1=...,x2=... and in a general solution form (vertical vector)

    Traditionally written solution: General solution form:

15. Give the definition of a vector space.

    Vector space is a set V with 2 operations: + (vector addition) * (scalar multiplication) The operation + (vector addition) must satisfy: (1) Commutative: For all vectors u and v in V, u + v = v + u(2) Associative: For all vectors u, v, w in V, u + (v + w) = (u + v) + w(3) Identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V, 0 + v = v and v + 0 = v(4) Inverse: For each vector v in V, the equations v + x = 0 and x + v = 0 have a solution x in V…

16. [Vector Space] What exactly is meant by vector addition and scalar multiplication?

    Vector addition and scalar multiplication are ANY operations satisfying the closure property: u + v is in V av is in V V - set of all vectors Some older sources mention these properties as separate axioms.

17. Apart from the set of actual vectors (n-dimensional (6, 6, 6, 54, ...)) - name 2 other things that can be considered a vector space.

    The set of all 2x3 matrices Also works: the set of all matrices of the SAME dimensionsThe set of all polynomials of degree 5 or less Also works: the set of all polynomials of degree N or less, or the set of all polynomials

18. Explain: why is the set of all matrices NOT a vector space? why is the set of all polynomials of degree 5 NOT a vector space?

    The set of all matrices: because there is no binary addition operation defined for matrices of different dimensions. (3, 6) + (9) = ???The set of all polynomials of degree 5: because v + v does NOT produce a vector. E.g. (9 + 2x5) + (6x2 - 2x5) = NOT a polynomial of degree 5.

19. Why is a set of vectors { v1 } always linearly independent?

    Because given a set consisting of only one vector, you cannot find not-all-zero scalars a such that av1 = 0.

20. Why is the set { 0, v1, v2, v3 } linearly dependent?

    Because there exist such not-all-zero scalars a, that: a10 + a2v1 + a2v2 + a3v3 = 0, e.g. by making a1 equal to 25.

21. Prove: if r1v1 + r2v2 + rnvn = 0 for some not-all-zero scalars r, then { v1, v2, vn } is linearly dependent.

    Because if not all scalars are 0, - then there exists some vector that can be expressed as: vn = r1v1 + r2v2 / -rn

22. Prove: if { v1, v2, vn } is linearly dependent - there exist such not-all-zero r, that r1v1 + r2v2 + rnvn = 0.

    Because by definition of linearly dependent, v1 = rv1 + rvn rv2 + rvn - 1* v1 = 0.

23. Name 3 vectors that span R3.

    R3 = span({ (1, 0, 0), (0, 1, 0), (0, 0, 1) }) There are many other possibilities for spanning sets for R3.For example, { 7i, 3j, -9k } will also work.Any three vectors of R3 will work as long as they are not linearly dependent.

24. Name 27 vectors that fail to span R3.

    E.g. { (1, 0, 0), (2, 0, 0), (3, 0, 0), ~, (27, 0, 0) }. No linear combination of these vectors will ever give you (0, 0, 88).

25. Is one of these bottles excessive?

26. How to determine whether, for these 4 bottles: One of the bottles is unnecessary (the set of these bottles is linearly dependent) Enough bottles to create any drink?

    Linearly dependent - a set of vectors is linearly dependent if Ax = 0 has a nonzero solution. The vectors are the columns of ASpan - a set of rows spans the vector space if Ax = v always has a solution. The vectors are the columns of A

27. Name a shortcut for determining whether { v1, v2, v3 } are linearly dependent span R3

    linearly dependent - matrix has a nontrivial solution (not all solutions are 0), that is, at least one row is zero in rref. spans R3 - matrix doesn't have any row of zeros. The vectors are the columns of A

28. What's a basis for a vector space?

    Basis for a vector space is a set of vectors that both spans it and is linearly independent.

29. Will every set of 4 vectors in R4 be a basis for R4?

    No! E.g., { (1, 0, 0, 0), (5, 0, 0, 0), (-4, 0, 0, 0), (88, 0, 0, 0) }: no linear combination of these vectors will give you (0, 0, 0, 5) and these vectors fail to be linearly independent too.

30. Name example of a basis for the vector space 'all polynomials'.

    E.g. { 1, x, x2, x3, x4, ... }. Some vector spaces that we have seen do not have bases with a finite number of elements, e.g. the vector space of real-valued functions or the vector space of all polynomials.We call those vector spaces infinite dimensional.

31. What's a subspace of a vector space Rn?

    W is a subspace of V if W is a subset of V, and W is closed under the same vector + and scalar * operations.

32. What's the smallest and the largest subspace of R2?

    Smallest: zero subspace: { (0, 0) }. Largest: the vector space R2 itself. Set that contains no vectors at all is not a subspace, because a definition of the vector space includes the word 'nonempty':Vector space is a nonempty set with a binary operation of vector addition, together with a binary operation of scalar multiplication.

33. What's dimension of a vector space?

    The dimension of a vector space is the number of vectors in any basis for the space. I.e. the number of coordinates necessary to specify any vector.

34. Name all 3 subspaces of R2.

    Zero subspace { (0, 0) } R2 itself Any line through the origin Nonempty set W is a subspace of V if W is closed under vector addition and scalar multiplication.