linear algebra

Question 1

calculating dot product through magnitude of vectors and cosine Theta

Answer 1

Question 2

calculating dot product through distances on the plane

Answer 2

Question 3

what is the dot product equal to, when to vectors are at a right angle to each other (90 degree)?

Answer 3

their dot product would always be zero

Question 4

why cosine Theta in the dot product?

Answer 4

to make vector “look in the same direction”, as if measuring the shadow of one of them lying on top of the other

Question 5

difference between vector and scalar?

Answer 5

Vectors have magnitude and direction

Scalars have magnitude only

Question 6

calculating the cross product with sinus Theta

Answer 6

Question 7

calculating the cross product through magnitudes

Answer 7

Question 8

main difference between dot product and cross product

Answer 8

the result of dot product is scalar quantity

the result of cross product is vector quantity

Question 9

What is the dot product of two orthogonal vectors equal to?

Answer 9

zero, because cos(90 degree) is equal to zero

Question 10

what does a dot product return?

Answer 10

magnitude (scalar)

Question 11

what does a cross product return?

Answer 11

another vector, perpendicular to the previous 2

Question 12

what does the magnitude of vector “c” designate, in relation to vectors “a” and “b”?

Answer 12

Question 13

what is cross product parallelogram?

Answer 13

Think of the parallelogram formed by vector a and vector b. The base of this parallelogram has length ∥a∥ (vector magnitude) and the height has length ∥b∥sin(θ). That means the area of the parallelogram in total is precisely the magnitude of the cross product.

Question 14

Right hand rule for cross product

Answer 14

Question 15

what do dot and cross products measure?

Answer 15

The dot product measures how much two vectors point in the same direction, but the cross product measures how much two vectors point in different directions.

Question 16

euclidian distance between two matrices (formula)

Answer 16

through Pythagoras, for any number of dimensions.

Question 17

what are the implicit dimensions of a zero matrix?

Answer 17

It is understood that the dimensions of the zero matrix match the dimensions of matrix A (the one it is being manimulated with.)

Question 18

5 properties of matrices

Answer 18

Question 19

When is multiplication of 2 matrices defined? (condition)

Answer 19

in order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix

Question 20

What is the multiplication product of m x n and n x k matrices?

Answer 20

m x k matrix

btw this multiplication of m x n and n x k is defined, because n1 is equal to n2, i.e. the number of columns of the first matrix is equal to the number of rows in the second matrix.

Question 21

Identity Matrix formula

Answer 21

A I = I A = A

Question 22

Identity Matrix form

Answer 22

Question 23

Identity Matrix dimensions

Answer 23

it is always a square, regardless of the other matrix´s shape

Question 24

properties of matrix multiplication

Answer 24

Question 25

what are collinear vectors?

Answer 25

vectors are collinear, if they lie on the same line or parallel lines.

Question 26

secant line?

Answer 26

сессанта

Question 27

Elementary row operations (Gauss-Jordan Elimination)

Answer 27

1. Swap the positions of two of the rows
2. Multiply one of the rows by a nonzero scalar.
3. Add or subtract the scalar multiple of one row to another row.