Let A=( 1 -1 0 2 2 -2 0 4 -1 1 1 -1 0

step 1 We're doing a vector W and R4. W has components 1, negative 2, negative 1 and 3. In part A, we'r

Let A=( 1 -1 0 2 2 -2 0 4...

Let $A=\left(\begin{array}{rrrr}1 & -1 & 0 & 2 \\ 2 & -2 & 0 & 4 \\ -1 & 1 & 1 & -1 \\ 0 & 0 & 2 & 2\end{array}\right)$. (a) Find an orthogonal basis for coimg $A$. (b) Find an orthogonal basis for ker $A$. (c) If you combine your bases from parts (a) and (b), do you get an orthogonal basis of $\mathbb{R}^4$ ? Why or why not?

Let $A=\left(\begin{array}{rrrr}1 & -1 & 0 & 2 \\ 2 & -2 & 0 & 4 \\ -1 & 1 & 1 & -1 \\ 0 & 0 & 2 & 2\end{array}\right)$.
(a) Find an orthogonal basis for coimg $A$.
(b) Find an
orthogonal basis for ker $A$. (c) If you combine your bases from parts (a) and (b), do you get an orthogonal basis of $\mathbb{R}^4$ ? Why or why not?

Key Concepts

Direct Sum Decomposition and Orthogonal Complements

Direct sum decomposition is the notion that a vector space can be written as a sum of two (or more) subspaces in such a way that every vector in the space can be uniquely expressed as a sum of vectors from each subspace. When these subspaces are chosen as a kernel and its orthogonal complement (or coimage), they form an orthogonal decomposition of the space, meaning that any vector from one subspace is orthogonal to any vector from the other.

Coimage

The coimage, in the context of linear transformations, refers to a subspace of the domain that complements the kernel. Often taken as the orthogonal complement of the kernel in an inner product space, it captures the part of the domain that is 'visible' under the transformation. Its dimension equals the rank of the matrix.

Gram-Schmidt Orthogonalization

The Gram-Schmidt process is a method for converting a set of linearly independent vectors into an orthogonal (or orthonormal) set that spans the same subspace. It is a fundamental tool for constructing orthogonal bases, which are advantageous for simplifying many problems in linear algebra.

Orthogonal Basis

An orthogonal basis is a set of basis vectors for a subspace in which every pair of distinct vectors is orthogonal with respect to a given inner product. Such a basis greatly simplifies computations in linear algebra, including projections, decompositions, and coordinate computations within the subspace.

Kernel (Null Space) of a Matrix

The kernel (or null space) of a matrix A is the collection of all vectors that are mapped by A to the zero vector. It is a subspace of the domain of A and represents the solutions to the homogeneous system A? = 0, indicating how the transformation induced by A fails to be injective.

Transcript

00:04 We're doing a vector w and r4.

00:08 W has components 1, negative 2, negative 1 and 3.

00:18 In part a, we're asked to find a orthogonal basis for w perp.

00:26 Well, with the vector v in r4, we'll say that components x, y, z, t.

00:35 If this is in w perth, this implies the inner product of v or wp.

00:40 Curve, let's not be kirk, put this w, that's be equal to zero.

00:51 So we have x minus 2y minus z plus 3 t equal 0.

01:05 We have a lot of freedom here when choosing basis factors.

01:15 3 degrees of 3.

01:16 We need precise.

01:18 Let's take x and y to be equal to 0 and 2 to be equal to 1.

01:23 Then it follows that v is equal to three.

01:28 And so we get the vector u1 with coordinates 0 -031.

01:48 Now we want to find an orthogma basis for w -p.

01:52 So we want to find another vector b, such that v is not only in a part of v with w -p, w is to 0, but also being a product of v with u1 has to equal to 0.

02:10 So now we get the equation when we had the 4, x minus 2y minus z plus 3t equal 0.

02:26 And we also have the equation 3z plus 2 equals 0.

02:31 We can simplify this system.

02:41 I will, well now we have two degrees of freedom.

02:48 So i'll take, uh, michigan j fraud, well, we get the new system, 3x minus 6 y, plus 10 t equals 0.

03:25 From the seinfeld graber corporation...

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