Understanding Null Space and Basis Vectors

The video tutorial explains how to find the basis for the null space of a matrix by solving the equation A * x = 0. It involves setting up the equation, performing row reduction, identifying free variables, and parameterizing the solution. The basis vectors are determined, and the values of i and j are found by matching the vectors to the given form.

What is the null space of a matrix?

• The set of all solutions to the equation Ax = 0

• The set of all non-zero vectors

• The set of all solutions to the equation Ax = b

• The set of all vectors orthogonal to the matrix

What is the dimension of vector x if matrix A is 2x3?

• 1 component

• 3 components

• 2 components

• 4 components

What is the result of multiplying a 2x3 matrix with a 3x1 vector?

• A 3x1 matrix

• A 2x1 matrix

• A 3x3 matrix

• A 2x3 matrix

What is the purpose of row operations in solving the equation Ax = 0?

• To simplify the matrix and identify basic and free variables

• To find the determinant of the matrix

• To transpose the matrix

• To find the inverse of the matrix

Which variables are considered free in the given matrix?

• x1 and x2

• x2 and x3

• x1 only

• x1 and x3

How is the solution to the equation Ax = 0 parameterized?

• By setting x2 = s and x3 = t

• By setting x1 = s and x2 = t

• By setting x1 = s and x3 = t

• By setting x1 = t and x3 = s

What is the vector formed by the coefficients of s in the parameterized solution?

• (-5, 1, 0)

• (2, 0, 1)

• (1, 0, 0)

• (0, 1, 0)

What is the vector formed by the coefficients of t in the parameterized solution?

• (0, 1, 0)

• (-5, 1, 0)

• (2, 0, 1)

• (1, 0, 0)

What are the basis vectors for the null space of matrix A?

• (-5, 1, 0) and (2, 0, 1)

• (1, 0, 0) and (0, 1, 0)

• (0, 0, 1) and (1, 1, 1)

• (2, 1, 0) and (-5, 0, 1)

What are the values of i and j in the vectors i(1, 0) and j(0, 1)?

• i = 0, j = 1

• i = 1, j = 0

• i = -5, j = 2

• i = 2, j = -5