Linear Algebra: Step By Step

52 flashcards · Mathematics · by lakesare

1. When can we define matrix powers An?

   For square matrices only.

2. What's A0?

   I.

3. Why (A + B)2 ≠ A2 + 2AB + B2?

   Because (A + B)2 = A2 + AB + BA + B2.

4. Show that matrix A doesn't have an inverse.

5. Show graphically what solutions are possible to a linear system with 2 unknowns.

   The graphs in Fig. 1.6 represent the three possible solutions to a linear system with two unknowns.

6. Show graphically what solutions are possible to a linear system with 3 unknowns.

   (a) shows three planes (equations) which have no point in common, hence no solution.(b) shows the three planes (equations) with a unique point in common.(c) shows three planes (equations) with a line in common. Every point on this line is a solution, which means we have an infinite number of solutions.

7. Let the top of a table be given by the coordinates P(1, 2), Q(2, 2), R(2, 4) and S(1, 4). Write down the matrix A which represents this table top PQRS.

8. Determine the image of the matrix F under the transformation AF.

   Applying transformation A italicizes the given letter F.

9. Show that matrices A and B are inverse of each other.

   We need to show that the matrix multiplications AB and BA give the identity matrix I. Similarly, we have BA = I. Therefore, matrices A and B are the inverse of each other.

10. Graphically show when a 3 by 3 linear system situations where has no solutions.

    In each of these cases there is no point which is common to all three planes, and this means that there is no solution to the linear system.

11. Why do we want to find the inverse matrix, when Gaussian elimination is often faster & easier?

    Bc in real life it's frequent to have a situation where b changes in Ax = b. There are many real life applications where you might need to solve the linear system, Ax = b, in which matrix A remains the same while vector b changes. In that case, it is more efficient to solve for the inverse of A once, then multiply that inverse by the various b vectors. This is because matrix A is generally a common property but vector b a particular property of the system.

12. What's an elementary matrix?

    An elementary matrix is a matrix obtained by a single row operation on the identity matrix I.

13. Which of the following are elementary matrices?

    A, B and D.

14. What effect does the matrix multiplication EA have on the general 3 by 3 matrix A?

    The elementary matrix E in this case is obtained from I by multiplying the middle row by −2. Carrying out the matrix multiplication EA gives: Hence the matrix multiplication EA performs the same elementary row operation of multiplying the middle row by −2. In general, if E is a n by n elementary matrix obtained by performing a particular row operation on the identity matrix In and A is an n by r matrix then the matrix multiplication EA performs the same row operation on A. We can use matrix E t…

15. A matrix B is row equivalent to a matrix A if and only if there are a finite number of elementary matrices E1, E2, E3, ... and En such that B = EnEn−1 ··· E2E1A.

    Matrices that are obtained from one another by a finite number of elementary row operations are said to be row equivalent. This definition means that the matrix B can be obtained from matrix A by a finite number of row operations. Note, that A is pre-multiplied (left multiplied) by elementary matrices. Note, that if matrix B is row equivalent to matrix A then the reverse is also true, matrix A is row equivalent to matrix B.

16. Why is doing EA equivalent to performing the same row operation [used to produce E] on A? I -> operation -> E EA results in same operation on A. If an elementary matrix, E, is obtained from the identity matrix by a certain row operation then the matrix multiplication EA produces the same operation on the matrix A.

    E (elementary matrix) is defined as a matrix obtained by a single row operation on I. Which means that E = SI, where S does this 'row operation' on I. EA = SIA = SA. Which means we do the same to A what we just did to A!

17. What's the definition for invertible matrix (non-singular)?

    Matrix is invertible if there exists matrix B such that AB = BA = I.

18. Why are invertible matrices always square?

    Matrix is invertible if there exists matrix B such that AB = BA = I, which means this matrix has to be multipliable by B from both sides.

19. Prove a → b. Let A be a n by n matrix, then the following 4 statements are equivalent: (a) The matrix A is invertible (non-singular). (b) The linear system Ax = O only has the trivial solution x = O. (c) The reduced row echelon form of the matrix A is the identity matrix I. (d) A is a product of elementary matrices.

20. Solve both linear systems. We know the inverse for this matrix:

21. Show that the matrix A is non-invertible.

24. What does it mean for vectors to be called 'orthonormal'?

    Orthonormal vectors have two properties: 1. All the vectors are orthogonal to each other (perpendicular to each other). 2. All vectors are normalized, that is they have a norm or length of 1 (unit vectors). Actually these are examples of perpendicular unit vectors called orthonormal vectors, which means that they are normalized and they are orthogonal.

25. Prove that this linear combination is unique. e is a standard unit vector, - meaning these are linearly independent.yk is a scalar.

26. Let v1,...,vm be vectors in Rn. If n < m - then vectors are linearly dependent.

27. Determine whether the following vectors are linearly dependent in R3.

28. Let u, v and w be any vectors in Rn. Prove that the vectors u + v, v + w and u − w are linearly dependent.

29. Let e1 and e2 be the standard unit vectors in R2 (meaning they are linearly independent). Prove that e1 and e1 + e2 are linearly independent.

30. [genius one]

31. Prove the following: If an invertible matrix A has a left inverse B and a right inverse C, then B = C.

32. What's a symmetric matrix?

    Symmetric matrix are matrix A such that AT = A, that is A transposed is equal to A.

33. Determine whether the following vectors span R2:

34. Determine whether the following vectors form a basis for R2:

35. Show that the set of 2 by 2 non-invertible matrices do not form a vector space.

36. Show that P(t), the set of polynomials of degree n or less, is indeed a vector space.

    We need to check all 10 axioms with the elements or vectors (polynomials) of this space which have the general polynomial form. 1. Closed under vector addition 2. Commutative law p(t) + q(t) = q(t) + p(t)3. Associative law p(t) + q(t) + r(t) = p(t) + q(t) + r(t) 4. Identity The zero polynomial O is the real number 0 and is in the set P(t).p(t) + O = p(t)5. Inverse For the polynomial p(t) we have the inverse as −p(t) andp(t) − p(t) = O6. Closed under scalar multiplication 7. Associative law k1(k…

37. Subset vs subspace?

    Subset - specific set of elements chosen from V. Subspace - subset that satisfies 10 axioms with respect to the same vector addition and scalar multiplication as V. Note the difference between subspace and subset. A subset is merely a specific set of elements chosen from V. A subset must also satisfy the 10 axioms of vector space to be called a subspace.

38. What's a subspace of vector space V?

    Subspace - subset of V that satisfies 10 axioms with respect to the same vector + and scalar * as V. A non-empty subset S of a vector space V is called a subspace of V if it is also a vector space with respect to the same vector addition and scalar multiplication as V.

40. Prove that in order to know whether subset S is a subspace of the vector space V you need to only show: that V is closed under vector + that V is closed under scalar *

    Required to prove that all 10 axioms of the last section are satisfied if we have our closures. 1. Closed under vector + 2. Commutative law 3. Associative law 4. Identity v + O = vWe can get 0 by doing 0v = O (since we're closed under scalar *)5. Inverse v − v = OWe can get -v by doing (-1)v = -v (since we're closed under scalar *)6. Closed under scalar * 7. Associative law 8. Identity 9. Distributive 10. Distributive Axioms in purple are satisfied because these axioms are true for all vectors…

42. Prove that ⤇ If subset S is a subspace of V - then ku + cv is also in S. ⤆ If ku + cv is in S - then subset S is a subspace of V. For any vectors u and v in S, and for any scalars k and c.

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

    The number of vectors in a basis of a non-zero vector space V is the dimension of the space. Why does the above definition say non-zero space? Because the zero vector space {O} is linearly dependent, there are no vectors in the basis of {O}, which means that it has dimension 0. (The zero vector does not need any axes.)

45. What is dim(M22) equal to?

46. What is dim(P2) equal to?

    dim(P2) = 3 Remember that the standard basis for P2 (the set of all polynomials of degree 2 or less) is the set {1, t, t2}, which means dim(P2) = 3 since the basis consists of three vectors. The dimension does not need to be the number of vectors in the standard basis; it can be any basis for the given vector space because the number of vectors is the same, as the following theorem shows. Note: a general misconception is that the dimension of a space is how many entries are required to specify…

50. Size of a basis is a sweet spot between _______ and _______?

    Between max amount of linearly independent vectors, and min amount of spanning vectors.

51. What are row and column vectors of B?

    Row vectors: Column vectors:

52. What's the row space of B?

    It is the space spanned by B's row vectors: Where k1, k2, k3 are scalars.