Determine the null space of the given matrix A. A=[ 1 -2 1 4 -7 -2

step 1 We are asked to find the null space of A. The definition of the null space is given at the top h

Determine the null space of the given matrix A. A=[ 1 ...

Key Concepts

Basis of a Subspace

The null space is not just any set, but a subspace of the domain of the matrix, and it is commonly described by its basis. A basis is a set of linearly independent vectors that span the subspace, providing a minimal and complete description of all solutions to the homogeneous system.

Gaussian Elimination

Gaussian elimination is a method often used to simplify a system of linear equations by converting the matrix into a row echelon form or reduced row echelon form. This process helps in identifying pivot and free variables, which are essential for expressing the null space as a span of basis vectors.

Dimension (Nullity)

The nullity of a matrix is defined as the number of free variables in the homogeneous system or, equivalently, the number of vectors in a basis for the null space. It represents the dimension of the null space and gives insights into the linear dependence among the columns of the matrix, connecting directly to the rank-nullity theorem.

Null Space

The null space of a matrix refers to the set of all vectors that, when multiplied by the matrix, yield the zero vector. It represents the solution set to the homogeneous equation Ax = 0 and is a fundamental concept in linear algebra because it provides information about the kernel of a linear transformation associated with the matrix.

Homogeneous System

Determining the null space is equivalent to solving a homogeneous system of linear equations, where the constant terms are all zero. This system always has at least the trivial solution (the zero vector), and the analysis of this system helps identify additional solutions when they exist, revealing important properties about the matrix's structure.

Transcript

00:01 We are asked to find the null space of a.

00:03 The definition of the null space is given at the top here with the null space of an n -by -n matrix is given by the set of vectors x in rn, such that the matrix a times the vector x is equal to the zero vector.

00:19 The problem gives the matrix a as a 3 -by -3 matrix shown here.

00:25 We can see from there that the value of n is 3, which means we're in r3.

00:30 And so we can write our x vector as a 3 by 1 matrix with x1 x2 and x3 so now we can start to solve this linear equation by using row reduction to get a in row echelon form so i've rewritten a in augmented form down here and we can start our row reduction by seeing that we can get we can get a zero here if we multiply minus 4 times the first row and add that to the second row.

01:25 So we're going to add minus 4 times row 1 and add it to 2.

01:40 So that gives our first row will stay the same.

01:44 1 minus 2 1 and 0.

01:51 And so we'll have a minus 4 times 1 added to 4 which gives 0.

01:57 We have a minus 4 times a minus 2 plus a minus 7 which gives 1.

02:04 We have minus 4 times 1 plus a minus 2 which is minus 6.

02:10 And our third row stays the same, minus 1, 3, 4.

02:15 And then we have a minus 4 times 0, which is 0, and the 0 of the 3rd row.

02:27 So now we can get a 0 here if we add 1 times row 1 to row 3.

02:39 So we'll add 1 times row 1 to row 3.

02:52 So that will give our first row will stay the same, and our second row will stay the same.

03:04 And then our third row will be 1 plus and minus 1, which is 0.