Find the closest point to 𝐛=(1,2,-1,3)^T in the subspace W=span{(1,0,2,1)^T., .(1,1,0,-1)^T,(2,0

Find the closest point to 𝐛=(1,2,-1,3)^T in the subspace...

Key Concepts

Orthogonal Projection

The orthogonal projection of a vector onto a subspace is the vector in the subspace that is closest to the given vector in terms of Euclidean distance. It is computed using the components along each basis vector of an orthogonal (or orthonormal) basis of the subspace. This projection minimizes the distance (or error) between the given vector and the subspace, and is determined by summing the projections onto all the basis vectors.

Gram-Schmidt Process

The Gram-Schmidt process is an algorithm for converting a set of linearly independent vectors into an orthogonal (or orthonormal) set that spans the same subspace. This process is essential when the given spanning set is not orthogonal, as it allows one to systematically remove the non-orthogonality to produce a basis that facilitates easier computation of projections and other vector operations.

Orthogonal Basis

An orthogonal basis for a subspace is a set of basis vectors that are mutually perpendicular. This property simplifies many calculations, especially projections, because the projection of a vector onto each basis element can be computed independently without the need to account for cross-terms. Using an orthogonal basis is crucial to simplify and accurately compute the components of a vector along the directions defined by the basis.

Subspace

A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. In problems like this, the subspace is defined as the span of a set of vectors, meaning every element in the subspace can be written as a linear combination of those vectors. Understanding subspaces is fundamental when dealing with projections, as the closest point to a given vector is sought within a specified subspace.

Transcript

00:02 In this question, we want to find the orthogonal projection of this vector onto the subspace spanned by v1 and v2, given here.

00:19 As you probably know from the textbook, or any other linear algebra textbook, the orthogonal projection of a vector onto the column space of a matrix is given by a simple formula, which is a times a transpose a inverse times a transpose u, where u is the vector you want to project and a is the column space of the matrix whose column space you want to project onto.

01:00 So since we want to project onto the span of these two vectors, we should create a matrix a, which has those two vectors as columns, because the column space of a matrix is just the span of its columns.

01:15 So now let's compute that formula for this projection, that is.

01:34 So first of all, let's calculate a transpose u.

01:40 That's the product i've written here in green.

01:44 The first entry will be the dot product of the first row of a transpose with u, so minus 1 minus 12 plus 1, and the second entry will be 2 2 4 dotted with 1 minus 6 1.

02:03 2 plus 4 minus 12.

02:05 Okay, so now we're one step there.

02:18 Notice that i factored the common scalar out of the vector to make the multiplication with these matrices and vectors more simple, simpler.

02:27 Now let's compute a transpose a.

02:35 So since a has two columns and a transpose has two rows, this will be a 2 by 2 matrix, and by definition of matrix multiplication, the diagonal entries will be the dot products of each column with itself, and the non -diagonal terms will be the dot product of the two columns together.

03:02 So the first diagonal entry is 1 plus 4 plus 1, the second is 4 plus 4 plus 16, and the off -diagonal terms are minus 2 plus 4 plus 4...

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