Determine the null space of A and verify the Rank-Nullity Theorem. A=[ 6 -1 9 ]

Name: Problem 1, Chapter 4: Differential Equations and Linear Algebra
Uploaded: 2020-05-24T18:01:03-08:00
Duration: 3 min 52 s
Channel: Patrick Vaughan
Description: Determine the null space of A and verify the Rank-Nullity Theorem. A=[ 6 -1 9 ]

Determine the null space of A and verify the Rank-Nullity...

Key Concepts

Rank-Nullity Theorem

The Rank-Nullity Theorem is a fundamental result in linear algebra that relates the dimensions of a linear transformation's domain, its null space, and its image (or rank). Specifically, for any linear transformation from a finite-dimensional vector space to another, the sum of the rank (the dimension of the image) and the nullity (the dimension of the null space) equals the dimension of the domain. This theorem is crucial for verifying the consistency and structure of solutions to linear equations.

Matrix Rank

The rank of a matrix is the dimension of the column space (or row space) and represents the number of linearly independent columns (or rows) in the matrix. It provides insights into the span of the matrix in its codomain and is fundamental in understanding the behavior of the corresponding linear transformation. The rank indicates how many dimensions of the input space are 'effectively used' by the transformation.

Null Space

The null space of a matrix (or linear transformation) is the set of all vectors that, when multiplied by the matrix, yield the zero vector. It forms a subspace of the domain of the matrix and is crucial for understanding solutions to homogeneous linear systems. Determining the null space reveals the degrees of freedom in the solution space, indicating how many independent parameters can be chosen without affecting the equality to zero.

Transcript

0:00 Hello.

00:01 Today we're going to find the null space of some matrix a and then verify the rank nullity serum.

00:06 So if we have some arbitrary matrix a equal to 1, negative 1, and 9, first we want to put this in reduced row echelon form.

00:19 So to do this, we already have our leading one or our pivot point at the top, but we need to zero out the entries below it.

00:29 So we'll do, we'll keep the first and we'll do row 2 plus row 1 so negative 1 plus 1 is 0 and then to 0 out bottom we'll just do row 3 plus negative 9 times row 1 which gives us 0 and this is reduced row echelon form so since we only have one column and one pivot point that tells us right away that the rank of this matrix a or vector a really equals 1.

01:08 Now let's continue on and solve for the null space.

01:12 So we have our reduced row atchuan form, 1 ,00, which is a three row by one column matrix, and then we want to multiply this by x1, which is a 1 by 1 by 1 matrix, which equals a 3x1 matrix again, of 0, 0, and 0.

01:41 So putting this into a systems of equations, we have the row by the column, and we have then x1 equals 0, 1 x1 equals 0, and then we have 0, sorry about that, still getting used to the platform, and then we have 0 equals 0 and 0 equals 0.

02:14 So putting this into the vector form, we have x1 equals some vector times x1, and we know that x1 equals x1 equals 0 from our system right here.

02:34 Underline that red.

02:36 So x1 equals 0 x1, which tells us, and that is our null space...

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