Linear Algebra: Matrices

Linear algebra is the study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering. Matrices, sets, tuples.

25 flashcards · Mathematics · by lakesare

  1. Multiply matrices.

    Each entry is the result of dot product of A's row and B's column, set in exactly that place. Humans have found defining the matrix multiplication the way it's defined now to have the most applications. It's a human construct.

  2. When is matrix multiplication defined?

    When we have the same number of rows/columns in the middle: A (m × K) and B (K × n)

  3. Calculate the dot product: (1, 2, 3) • (7, 9, 11)

    (1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11 = 58

  4. What does equal to?

    = 12 + 22 + 32 + 42 = 30 It's kind of like a loop: for(let i = 1; n !== 4 + 1; n++){ n2 }

  5. What's an identity matrix?

    n x n with 1s in the diagonal

  6. Why did matrices evolve?

    Matrices evolved in order to solve linear equations. Basically it's a syntax sugar over them.

  7. In what form a solution to a system of linear equations is usually given?

    Column vector: (x, y, z from top down).

  8. What is a tuple, how is it denoted?

    Tuple is a finite ordered list (a la array). (2, 4, 1, 7)

  9. Give an example of 2-tuple.

    (78, -4)

  10. How many distinct 4-bit values are there?

    0111, 1111, ... 2^4 = 16.

  11. Name 2 De Morgan's laws.

    !(A || B) == !A && !B; !(A && B) == !A || !B;

  12. When is matrix in a row echelon form?

    The first non-zero element in each row, called the leading entry, is 1. Each leading entry is in a column to the right of the leading entry in the previous row. Rows with all zero elements, if any, are below rows having a non-zero element.

  13. When is matrix in reduced row echelon form?

    The matrix is in row echelon form (i.e., it satisfies the three conditions listed above). The leading entry in each row is the only non-zero entry in its column.

  14. Name 3 matrix operations that are allowed.

    Replace row (by ADDING one row to another) Switch places of two rows Multiply a row 1. Add multiples of one row to another row.2. Interchange two rows.3. Multiply a row by a non-zero real number.

  15. What's a leading entry in a matrix?

    First nonzero entry in every row. Let M be an m × n matrix. Then an entry of M is said to be a leading entry if it is the first non-zero entry in some row.

  16. Name 3 results Gaussian Elimination can lead to. To perform Gaussian Elimination on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations:- Swapping two rows,- Multiplying a row by a nonzero number,- Adding a multiple of one row to another row.

    Case 1: No solution Some leading entry occurs in the final column b ∗ . In this case we deduce that there is no solution. For example: Case 2: Single solution The final column does not contain a leading entry but all other columns do contain a leading entry. Case 3: Infinite amount of solutions The final column and some other columns do not contain leading entries.

  17. Multiply matrices:

  18. Given a square matrix: How would you compute A3?

    A3 = AA2

  19. How to compute A4 power of a matrix?

    A4 = AA3 A3 = AA2 A2 = AA (usual matrix multiplication here)

  20. What matrix is denoted by I5?

    A 5x5 square identity matrix. Identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros.The effect of multiplying a given matrix by an identity matrix is to leave the given matrix unchanged.

  21. What's a great property of identity matrices?

    When A is m×n, it is a property of matrix multiplication that

  22. Associativity VS commutativity?

    associative operation: calculation that gives the same result regardless of the way the numbers are grouped. commutative operation: changing the order of the operands does not change the result.

  23. What is this property called?

    Associativity In mathematics, an associative operation is a calculation that gives the same result regardless of the way the numbers are grouped. Addition and multiplication are both associative, while subtraction and division are not.

  24. Matrix multiplication is ______________, but not ______________.

    Matrix multiplication is associative, but not commutative. Suppose that we have four matrices A, B, C and D of dimensions 2 × 1, 1 × 5, 5 × 2 and 2 × 3 respectively, denoted by A2×1, B1×5, C5×2, D2×3. We want to compute the multiplication ABCD.Since matrix multiplication is associative, we can conduct the multiplication between any two adjacent matrices prior to others. The final result will be the same. For example, we can multiply AB before multiplying C and finally D. That is, ((AB)C)D. We c…

  25. What is the number of multiplications and additions do we need to perform to multiply these matrices?