No Bullshit Guide To Linear Algebra

Following a great book about linear algebra by Ivan Savov: 'No Bullshit Guide To Linear Algebra' Visual: http://immersivemath.com/ila/ch04_vectorproduct/ch04.html In 4 pages: https://minireference.com/static/tutorials/linear_algebra_in_4_pages.pdf

66 flashcards · Mathematics · by lakesare

1. A function f() is linear when?

   A linear function must satisfy 2 properties: f(x + y) = f(x) + f(y) f(ax) = af(x) The prototypical example that gives linear maps their name is the function f : R → R : x ↦ cx, of which the graph is a line through the origin

2. What are 2 meanings of the term 'linear function'?

   In calculus: y = kx + b, a polynomial of degree one or less In linear algebra: a map f between two vector spaces that preserves vector addition and scalar multiplication. y = kx + b is NOT a linear function in the linear algebra sense!

3. Cross product VS cartesian product?

   cross product - a × b cross product of vectors a and b cartesian product - A × B cartesian product of sets A and B. Result is a set of all possible tuples of elements from both sets.

4. What does this symbol denote?

   A set of all 3-dimensional vectors.

5. Explain all of these vector operations: Addition Subtraction Scaling Dot product Cross product Length

   Since adding two vectors v and w corresponds to adding their components, you might logically think that the product of two vectors also corresponds to the product of their components.However, this type of product is not used. Instead, we’ll define the dot product and the cross product, which allow us to perform useful geometric operations with vectors.

6. Write down signatures for: vector addition/subtraction vector dot product vector cross product

   vector addition/subtraction operation - Rn × Rn → Rn vector dot product operation - Rn × Rn → R It is also known as the scalar product, since the result of the dot product is a scalar number—a number that does not change when the basis changes.vector cross product operation - R3 × R3 → R3 It is also sometimes called the vector product, since the output of this operation is a vector.The cross product of two vectors a and b is defined only in 3-dimensional space.

7. What's the geometrical meaning of vector dot product VS vector cross product?

   Dot product - a · b results in number that's equal to lengths of a and b, multiplied by the cos() between them. Cross product - a × b results in vector, whose: direction is perpendicular to both a and b. length is equal to:

8. What's the right-hand rule in relation to the cross product between two vectors a × b?

   We can use the right-hand rule to determine the direction of the vector resulting from a × b. When your index finger points in the same direction as the vector a middle finger points in the direction of vector b your thumb will point in the direction of a × b. Consider the plane formed by the vectors a and b. There are actually two vectors perpendicular to this plane: one above the plane and one below the plane.We use the right-hand rule to figure out which of these vectors corresponds to the c…

9. Is dot product commutative? Is cross product commutative?

   dot product is commutative: a · b = b · a cross product is anticommutative: a × b = - b × a It’s likely that, until now, the products you’ve seen in math have been commutative, which means the order of the inputs doesn’t matter.

10. How to calculate the cross product between vectors A and B?

    The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b.

11. Why is the dot product sometimes called scalar product, and cross product is sometimes called vector product?

    dot product is called scalar product, because it produces a number. Dot product is also known as the scalar product, since the result of the dot product is a scalar number - a number that does not change when the basis changes. The dot product is also sometimes called the inner product.cross product is called vector product, because it produces a vector. Cross product is also sometimes called the vector product, since the output of this operation is a vector.

12. What operation plays the role of division for matrices?

    Matrix inverse. The inverse matrix A-1 undoes whatever A did. The matrix inverse plays the role of "division by the matrix A" in matrix equations.

13. What's trace of the matrix?

    The trace of an n×n matrix is the sum of the n values on its diagonal aii.

14. What's the determinant of the 2×2 matrix?

    For the 3×3 matrix:

15. What does matrix determinant signify geometrically?

    Determinant of a matrix A tells you the volume of a box with sides given by rows of A. The determinant describes the relative geometry of the vectors that make up the rows of the matrix.

16. Name 3 multiplication-like operations used on vectors. Remember vector is a type of a matrix, and can be represented as either vertical or horizontal matrix.

    dot product cross product matrix multiplication

17. What's augmented matrix?

    The matrix corresponding to the system of equations, where we draw the vertical line to denote the constants past the equal sign. The first column corresponds to the coefficients of the first variable; the second column is for the second variable; and the last column corresponds to the constants on the right side. For the following system of equations:

18. What do REF and RREF stand for?

    REF - Row echelon form RREF - Reduced row echelon form

19. What's the difference between Gaussian elimination and Gauss-Jordan elimination?

    Gaussian elimination - process of bringing a matrix into row echelon form. Gauss-Jordan elimination - is the process of bringing a matrix into REDUCED row echelon form. The process of row reducing until the matrix is reduced is sometimes referred to as Gauss–Jordan elimination, to distinguish it from stopping after reaching echelon form.

20. What's the difference between REF and RREF? REF - Row echelon formRREF - Reduced row echelon form

    A matrix is said to be in row echelon form (REF) if all entries below the leading ENTRIES are zero. This form can be obtained by adding or subtracting the row with the leading one from the rows below it. A matrix is said to be in reduced row echelon form (RREF) if all the entries below and above the leading ONES are zero. Starting from the REF, we obtain the RREF by subtracting the row containing the leading ones (also called pivots) from the rows above that row.

21. Name 3 operations allowed in Gaussian/Gauss-Jordan elimination.

    Swapping two rows, (duh) Multiplying a row by a nonzero number, Adding a multiple of one row to another row.

22. Describe Gauss–Jordan elimination algorithm.

    The Gauss–Jordan elimination algorithm proceeds in two phases: a forward phase in which we move left to right, and a backward phase in which we move right to left. Forward phase (left to right): Obtain a pivot (a leading one) in the leftmost column. Subtract the row with the pivot from all rows below it to obtain zeros in the entire column. Look for a leading one in the next column and repeat. If there is no leading one - create it by multiplication. Results in Row Echelon Form Backward phase (…

23. What's the product of matrix and its inverse?

    Identity matrix.

24. How to solve for A in the matrix equation AB = C? How to solve for B in the matrix equation AB = C A, B, and C are matrices

25. Solve for X in the matrix equation: XA = B. You can assume the matrices A, B, C, D, and E are all invertible.

    X = BA-1

26. Solve for X in the matrix equation: AC = XDC. You can assume the matrices A, B, C, D, and E are all invertible.

    X = AD-1

27. Why multiplying matrices is not commutative? Give an answer in terms of linear transformations.

    Multiplying a vector by the matrix is akin to applying a linear transformation to that vector. g(f(x)) != f(g(x)), which means that AB != BA.

28. What does it mean for the set of vectors to be linearly independent?

    It means that none of the vectors in this set can be written as a linear combination of the other vectors. In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent.

29. When is matrix invertible?

    When its determinant is nonzero. det(A) != 0.

30. How to calculate the determinant of the 3×3 matrix?

    The formulas for the determinants of larger matrices are defined recursively. For example, the determinant of a 3×3 matrix is defined in terms of 2×2 determinants.

31. What's minor M24 of the matrix A?

    A "minor" is the determinant of the square matrix formed by deleting one row and one column from some larger square matrix.

32. Here is a general formula for the determinant of the n×n matrix: What does it assume, and how can we modify this formula?

    This determinant formula assumes that we expand the determinant along the first row of the matrix. However, we can find the determinant of the matrix by expanding any row or column!

33. Find the area of the parallelogram created by corner points: (0, 0) vector v vector w vector v + w

    Its area is equal to the determinant of the matrix, where first ROW is v second ROW is w The determinant of a 2×2 matrix corresponds to the area of the parallelogram constructed from the rows of the matrix.

34. What does the determinant of the 3×3 matrix represent geometrically?

    It corresponds to the volume of the parallelepiped constructed from the rows of the matrix.

35. Prove that dot product with itself (u ⋅ u) is equal to own length squared (|u|2).

36. Prove that: v ⋅ w = ‖v‖ ‖w‖ cos𝜃.

    1. Law of cosines 2. Algebraic properties of dot product (!!!) Combining these:

37. What does the SIGN of determinant of the 2×2 matrix represent?

    if D > 0, then angle is (0; π) if D < 0, then angle is (π; 2π) if D = 0, the angle is either 0 or π (vectors point in the same direction, or vectors point in the opposite directions)

38. What's Cavalieri's principle? Francesco Cavalieri 1598

    If, in two solids of equal altitude, the sections made by planes parallel to their bases, and at the same distance from their bases are always equal, then the volumes of the two solids are equal. Works in 2 dimensions with a line too. Cavalieri's principle was originally called the method of indivisibles, the name it was known by in Renaissance Europe.

39. Determinant of the 2×2 matrix gives us area of the parallelogram. Show why D = |v| |w| sinθ.

    The formula for the area of a parallelogram is A = bh Where b is the length of the parallelogram's base, and h is the parallelogram's height.

40. How does each of row operations used in the Gauss–Jordan elimination procedure affect matrix's determinant? Add a multiple of one row to another row Swap two rows Multiply a row by a constant

    Add a multiple of one row to another row This property follows from the fact that parallelepipeds with equal base enclosed between two parallel planes have the same volume even if they have different slants.This is known as Cavalieri's principle. It is easier to visualize Cavalieri's principle in two dimensions by considering two parallelograms with base b and different slants, enclosed between two parallel lines. The area of both parallelograms is the same A = b * h, where h is the distance be…

41. What's a rank of the matrix?

    The number of nonzero rows in the RREF.

42. When is matrix rank-deficient VS at full rank?

    n×n matrix is: at full rank - if its RREF contains n pivots rank-deficient - if it's RREF contains < n pivots (has a row of zeros)

43. How is determinant of the matrix connected to its rank?

    If det(A) = 0, then matrix A is linearly dependent, and has a row of zeros, which means it's rank-deficient. If det(A) != 0, we know that rref(A) = I (Identity matrix).

44. How are cross-product and determinant connected?

    We can compute cross-product of 2 vectors by computing the determinant of the special matrix.

45. What's Cramer's rule?

    In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.

46. Name 2 ways to find out whether matrix is linearly dependent.

    The matrix is linearly dependent if: Gauss-Jordan elimination - if there are any 0-only rows in RREF. Determinant - if determinant of the matrix is equal to 0.

47. Name 2 applications of the matrix determinant.

    Matrix is invertible when its determinant is nonzero. Determinant is the area of the parallelogram (or volume of parallelepiped).

48. What is a characteristic polynomial pA(λ) of a matrix?

    Determinant of the expression (A - λIn), where λ is a customarily used.

49. Why is pA(λ) called the characteristic polynomial of matrix A?

    Because pA(λ) encodes the information about 3 important properties of the matrix A: eigenvalues (λ1, λ2), trace Tr(A), and determinant det(A).

50. Find the determinant:

    det(A) = 2

51. Find the determinant:

    det(A) = 0 This matrix is also: linearly dependent has deficient rank

52. Find the volume of the parallelepiped whose sides are the vectors u = (1, 2, 3) v = (2, -2, 4) w = (2, 2, 5)

    V = 2.

53. Determine whether the set of vectors {(1, 4, 3), (2, 1, 1), (0, -2, -1)} is linearly dependent or linearly independent.

    Linearly independent, because det(A) != 0.

54. Name 2 methods to compute a matrix inverse A-1.

    1. Adjugate matrix approach 2. Using Gauss-Jordan elimination

55. Give the general formula for the determinant of the n×n matrix. What's minor? What's cofactor?

    Mij is called the minor associated with the entry aij.cij is called the cofactor, that is Mij * (-1)1+j.Here we can see where the name cofactor comes from: the cofactor cij is what multiplies the factor aij in the determinant formula.

56. What is adjugate matrix adj(A)?

    The adjugate matrix is the transpose of the matrix of cofactors C. Cofactor is a (-1)ij * Mij: The matrix of cofactors is: The matrix of cofactors is a matrix of the same dimensions as the original matrix A that is constructed by replacing each entry aij by its cofactor cij.

57. Give the general formula for the inverse of the matrix A-1 based on the adjugate matrix adj(A).

58. What does this Rn denote?

    The set of all possible n-dimensional vectors. A vector subspace consists of a subset of all possible vectors.

59. What is a span of a set of vectors?

    The span of a set of vectors - set of vectors that can be constructed as linear combinations of these vectors

60. What is a linear combination of vectors?

    Vector obtained by adding two or more vectors which are multiplied by scalar values. a is a linear combination of b and c.

61. Name 4 fundamental vector spaces associated with the matrix M.

    Row space - set of vectors that are all possible linear combinations of rows of M. Column space - set of vectors that are all possible linear combinations of columns of M. Null space - set of vectors v such that M v = 0 Left null space - The dimensions of the column space and the row space of a matrix are equal. We call this dimension the rank of the matrix: dim(C(M)) = dim(R(M)).

62. Name 2 useful properties of vector spaces. A vector space V consists of a set of vectors and all possible linear combinations of these vectors.

    Vector spaces is a set of vectors which are: closed under addition - for all vectors in that space, the sum of two vectors results in a vector in that vector space. closed under scalar multiplication - starting from any vector v in the vector space V and scaling the vector by any constant α results in a vector in that vector space. These two properties codify the essential nature of what a vector space is: a space of vectors that can be added together and scaled by constants.

63. How to go from parametric equation to symmetric equation? from symmetric equation to parametric equation? Note that L is not a vector, - it's a set of many vectors satisfying a particular property.

    From parametric equation to symmetric equation: 1. Go from V vector from Vx, Vy representation.2. Get rid of t. From symmetric equation to parametric equation: 1. Assign t to some value2. Go from Vx, Vy representation to V vector.

64. Consider the plane that contains the points p = (1, 0, 0) q = (0, 1, 0), and r = (0, 0, 1). Find geometric equation, general equation, and parametric equation for this plane.

    https://sciencing.com/vector-perpendicular-8419773.html

65. Find the general plane equation Ax + By + Cz = D for a plane that passes through 3 points

    https://www.emathhelp.net/calculators/linear-algebra/reduced-row-echelon-form-rref-caclulator/?i=%5B%5B1%2C-2%2C1%2C-1%2C0%5D%2C%5B4%2C-2%2C-2%2C-1%2C0%5D%2C%5B4%2C1%2C4%2C-1%2C0%5D%5D&reduced=on&steps=on

66. What's the dimension of vector space V?

    Number of vectors in a basis of the vector space V. E.g., the dimension of ℝn is n.E.g., the dimension of the vector space of polynomials in x with real coefficients having degree at most two is 3. Let V be a vector space not of infinite dimension. An important result in linear algebra is the following: Every basis for V has the same number of vectors.