Determine the null space of A and verify the Rank-Nullity Theorem. A=[ 1 4 -1 3

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

Key Concepts

Null Space

The null space of a matrix is the set of all solution vectors that, when multiplied by the matrix, yield the zero vector. It represents the collection of all inputs that the associated linear transformation sends to zero, and it is a subspace of the domain. This concept is critical for understanding the structure of solutions to homogeneous systems of linear equations.

Rank-Nullity Theorem

The Rank-Nullity Theorem is a fundamental principle in linear algebra that relates the dimension of the null space (nullity) of a matrix to the dimension of its column space (rank) and the number of columns in the matrix. It states that the sum of the rank and the nullity of a matrix equals the total number of its columns, offering key insight into the balance between independent directions and the degrees of freedom remaining in the solution set.

Gaussian Elimination

Gaussian elimination is a systematic method used to simplify matrices into row echelon form (or reduced row echelon form) in order to solve systems of linear equations. This process is essential for determining the rank of a matrix as well as its null space, as it exposes the linear dependencies among the rows and clarifies the structure of the solution space.

Basis and Dimension

A basis of a vector space or subspace is a set of linearly independent vectors that span the entire space. In the context of the null space, finding a basis helps characterize all solutions to a homogeneous system by expressing each solution as a linear combination of the basis vectors. The number of vectors in the basis is called the dimension of the space, which is a key element in the Rank-Nullity Theorem.

Transcript

0:00 Hello.

00:01 Today we're going to find the null space of some matrix, and we're also going to verify the rank nullity theorem.

00:08 So our given matrix today is a equals 1, 4, negative 1, 3, 2, 9, negative 1, 7, and 2, 8, negative 2, and 6.

00:30 So we're going to want to put this into reduced row echelon form.

00:37 So right from the get go, we can see that we want to preserve the first row because we already have that leading one in this initial operation.

00:52 And we can actually do row two plus negative two times row one, which gives us zero, nine plus 4 times negative 2, negative 8, so that's 1.

01:11 And then negative 1 plus negative 1 times negative 2.

01:16 So negative 1 plus 2 is again 1.

01:19 And then 7 minus 6, which is again 1.

01:25 And then for the third row, we can actually also do row 3 plus negative 2 times row 1.

01:36 And as you see straight across the board, basically if you just multiply by negative 2, it cancels out everything in row 3.

01:43 So we're actually going to get 0, 0, 0, and 0.

01:49 So let's rewrite that again.

01:52 1, 4, negative 1, 3, 0, 1, 1, 1, and then all 0.

02:04 So we know that we are preserving the bottom row because that is all zeros, which is already in its most reduced form.

02:16 And we have a leading one and we have a leading one in this column as well.

02:22 So that's already good.

02:24 But we have, if we look here, we have a number over the one in the second row.

02:31 So then when i get rid of that...

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